#72307735

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#21891
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂	#382487
prime radical	#10575331	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4137 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:

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

Polycyclic presentation:

2^10|b^2=fj,c^2=fj,d^2=e,e^2=i,g^2=i,i^2=j,b^a=bf,d^a=dh,d^b=dg,d^c=dfi,e^b=ei,f^d=fj,g^b=gi,g^c=gj,h^b=hj,i^b=ij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307810, #72308411
maximal normal subgroup of	#72307810, #72308411

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