#72343227

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#393031
commutator	C₂×#131	C₂³×C₄
Frattini, cosocle	C₄×#131	C₂⁴
socle	C₂	#72343138
prime radical	#72217290	C₂
LCS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×#131 ◁ G
LECS	G ▷ C₄×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

2763 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:

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

Polycyclic presentation:

2^11|a^2=e,b^2=i,d^2=hj,e^2=i,h^2=j,i^2=k,j^2=k,b^a=bg,c^a=ck,c^b=cf,d^a=dj,d^b=dh,d^c=dijk,e^d=ek,f^a=fijk,f^d=fk,g^c=gij,g^f=gk,h^a=hk,h^b=hjk,j^b=jk

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343240
maximal normal subgroup of	#72343240

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