#72307334

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	#24089
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂	#363929
prime radical	#10576601	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#2932 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4221 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:

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

Polycyclic presentation:

2^10|a^2=f,b^2=f,d^2=eghij,e^2=j,g^2=i,i^2=j,b^a=bfj,c^a=cfj,d^a=de,d^b=dg,d^c=dfj,e^c=ej,f^d=fj,g^a=gj,g^b=gi,h^b=hj,i^b=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307537
maximal normal subgroup of	#72307537

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