#72225512

order	4096
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#72214525
commutator	C₄×C₈	C₂⁴×C₈
Frattini, cosocle	C₄²×C₈	C₂⁵
socle	C₂	#72216519
prime radical	#72225394	C₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₄×#346 ◁ G
LECS	G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

7982 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^12|a^2=l,c^2=f,d^2=ghjk,e^2=hk,f^2=i,g^2=jl,h^2=kl,j^2=l,k^2=l,c^a=cg,d^a=dg,d^b=dh,d^c=djk,e^b=eh,e^c=ek,f^a=fj,f^e=fl,g^a=gj,g^c=gl,h^b=hk,h^c=hl,i^a=il,j^a=jl,k^b=kl

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227231
maximal normal subgroup of	#72227231

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.6);;
gap> SetPower(coll, 4, F.7*F.8*F.10*F.11);;
gap> SetPower(coll, 5, F.8*F.11);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 7, F.10*F.12);;
gap> SetPower(coll, 8, F.11*F.12);;
gap> SetPower(coll, 10, F.12);;
gap> SetPower(coll, 11, F.12);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.10*F.11);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.12);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.12);;
gap> SetConjugate(coll, 8, 2, F.8*F.11);;
gap> SetConjugate(coll, 8, 3, F.8*F.12);;
gap> SetConjugate(coll, 9, 1, F.9*F.12);;
gap> SetConjugate(coll, 10, 1, F.10*F.12);;
gap> SetConjugate(coll, 11, 2, F.11*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);