#72319935

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#53832
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
socle	C₂	#3343287
prime radical	#7504666	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3336 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=f,c^2=gj,h^2=j,b^a=bh,c^a=cg,c^b=cfgh,d^a=dgi,d^b=dfgi,d^c=dfgh,e^b=eghij,e^d=eghj,f^a=fj,f^e=fj,g^b=gj,g^d=gj,h^e=hj,i^a=ij,i^b=ij,i^d=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320148
maximal normal subgroup of	#72320148

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