#72307287

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#20975
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂	#364394
prime radical	C₂×#24285	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#2929 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3791 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:

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

Polycyclic presentation:

2^10|b^2=e,c^2=i,d^2=egi,g^2=ij,i^2=j,b^a=bh,c^a=cf,c^b=cfij,d^a=df,d^b=dg,d^c=dfj,e^c=ej,f^b=fj,g^b=gi,g^c=gj,h^c=hj,h^d=hj,i^b=ij

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307299, #72343932, #72343980
maximal normal subgroup of	#72307299, #72343932, #72343980

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