#72258148

order	4096
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₄×#12067
commutator	C₂×C₁₆	C₂²×C₄×C₈
Frattini, cosocle	C₂×#1311	C₂⁴
socle	C₂	#72258147
prime radical	#33981	C₂×C₈
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₁₆ ◁ C₄×#346 ◁ #72225138 ◁ G
LECS	G ▷ C₂×#1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1790 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,7,15)(3,16,20,48)(4,9,23,40)(5,11,27,42)(6,14,30,13)(8,31,34,17)(10,28,43,24)(12,44,37,25)(18,52,54,64)(19,47,29,49)(21,32,50,61)(22,39,58,38)(26,45,53,63)(33,51,46,55)(35,56,62,57)(36,59,41,60);(1,3)(2,8)(4,21)(5,26)(6,29)(7,20)(9,16)(10,32)(11,35)(12,45)(13,46)(14,33)(15,34)(17,25)(18,24)(19,30)(22,57)(23,50)(27,53)(28,54)(31,44)(36,52)(37,63)(38,47)(39,49)(40,48)(41,64)(42,62)(43,61)(51,60)(55,59)(56,58);(2,9,11,37,14,39,43,36,15,40,42,12,13,38,10,41)(3,17,18,50,19,51,53,57,20,31,54,21,29,55,26,56)(4,22,23,58)(5,27)(8,32,33,35,34,61,46,62)(16,45,47,52,48,63,49,64)(24,28)(25,60,44,59);(2,10,13,42,15,43,14,11)(3,18,19,53,20,54,29,26)(9,36,38,37,40,41,39,12)(17,21,51,56,31,50,55,57)

Polycyclic presentation:

2^12|a^2=l,c^2=e,d^2=g,e^2=h,f^2=ikl,g^2=j,h^2=j,i^2=k,k^2=l,b^a=bf,c^a=cf,c^b=cfi,d^a=dikl,d^b=dikl,d^c=dj,e^a=ei,e^b=eikl,f^a=fi,f^b=fi,f^c=fk,f^e=fl,g^a=gkl,g^b=gkl,h^a=hk,h^b=hkl,i^a=ikl,i^b=ikl,i^c=il,j^a=jl,j^b=jl,k^a=kl,k^b=kl

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258194
maximal normal subgroup of	#72258194

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j", "k", "l");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.12);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.9*F.11*F.12);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.11);;
gap> SetPower(coll, 11, F.12);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.9*F.11*F.12);;
gap> SetConjugate(coll, 4, 2, F.4*F.9*F.11*F.12);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.9*F.11*F.12);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.12);;
gap> SetConjugate(coll, 7, 1, F.7*F.11*F.12);;
gap> SetConjugate(coll, 7, 2, F.7*F.11*F.12);;
gap> SetConjugate(coll, 8, 1, F.8*F.11);;
gap> SetConjugate(coll, 8, 2, F.8*F.11*F.12);;
gap> SetConjugate(coll, 9, 1, F.9*F.11*F.12);;
gap> SetConjugate(coll, 9, 2, F.9*F.11*F.12);;
gap> SetConjugate(coll, 9, 3, F.9*F.12);;
gap> SetConjugate(coll, 10, 1, F.10*F.12);;
gap> SetConjugate(coll, 10, 2, F.10*F.12);;
gap> SetConjugate(coll, 11, 1, F.11*F.12);;
gap> SetConjugate(coll, 11, 2, F.11*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);