#72284590

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, socle	C₂²	#46996
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	#245285	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2307 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^10|a^2=i,c^2=f,d^2=hij,e^2=i,b^a=bij,c^a=ci,c^b=ch,d^a=dhi,d^c=dg,e^a=ef,e^b=egi,e^d=efj,f^b=fj,f^d=fi,g^a=gj,g^c=gi,g^d=gj,h^c=hj,h^e=hi

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285517
maximal normal subgroup of	#72285517

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