#72285199

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
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2354 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)(3,8)(4,9)(5,17)(6,21)(7,11)(10,20)(12,22)(13,27)(14,24)(15,25)(16,26)(18,23)(19,28)(29,31)(30,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,53)(44,60)(45,52)(46,56)(47,55)(48,51)(49,57)(50,59)(54,62)(58,64)(61,63);(1,3)(2,8)(4,14,16,12)(5,18,20,13)(6,7)(9,24,26,22)(10,27,17,23)(11,21)(15,30)(19,29)(25,32)(28,31)(33,35)(34,39)(36,43,47,59)(37,48)(38,45)(40,53,55,50)(41,51)(42,52)(44,49)(46,61,62,58)(54,64,56,63)(57,60);(1,4)(2,9)(3,12)(5,19,20,15)(6,14)(7,16)(8,22)(10,28,17,25)(11,26)(13,29,18,30)(21,24)(23,31,27,32)(33,36,38,47)(34,40,42,55)(35,43)(37,46,49,62)(39,50)(41,56,57,54)(44,61)(45,59)(48,58)(51,63)(52,53)(60,64);(1,5,7,20)(2,10,11,17)(3,13,6,18)(4,15)(8,23,21,27)(9,25)(12,29)(14,30)(16,19)(22,31)(24,32)(26,28)(33,37)(34,41)(35,44,45,48)(36,46,47,62)(38,49)(39,51,52,60)(40,54,55,56)(42,57)(43,58)(50,63)(53,64)(59,61);(1,6,7,3)(2,8,11,21)(4,12,16,14)(5,18,20,13)(9,24,26,22)(10,23,17,27)(15,29,19,30)(25,32,28,31)(35,45)(36,47)(37,49)(39,52)(40,55)(41,57)(43,59)(50,53)

Polycyclic presentation:

2^10|b^2=f,c^2=fhj,d^2=hij,e^2=ij,c^a=cfh,c^b=cg,d^a=dhij,d^b=dgh,d^c=dfgi,e^a=eij,e^b=efi,e^c=efhi,e^d=ej,f^c=fi,f^d=fij,g^a=gj,g^b=gi,g^c=gj,g^d=gj,g^e=gij,h^b=hj,h^c=hi

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286185
maximal normal subgroup of	#72286185

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