#72285615

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	#245279	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2249 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:

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

Polycyclic presentation:

2^10|a^2=ij,b^2=ij,c^2=gh,e^2=ghj,b^a=bij,c^a=cf,c^b=cg,d^a=dfij,d^b=dfj,d^c=dfhi,e^a=efgj,e^b=efj,e^c=eghj,e^d=ehij,f^c=fi,f^e=fij,g^c=gij,g^d=gj,g^e=gj,h^a=hi,h^b=hij,h^c=hij,h^e=hj

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285772
maximal normal subgroup of	#72285772

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