#72284776

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

2356 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)(3,7)(4,8)(5,17)(6,10)(9,18)(11,25)(12,27)(13,20)(14,19)(15,22)(16,24)(21,28)(23,30)(26,31)(29,32)(33,34)(35,39)(36,40)(37,50)(38,43)(41,49)(42,53)(44,57)(45,54)(46,60)(47,56)(48,58)(51,55)(52,59)(61,64)(62,63);(1,3)(2,7)(4,14)(5,12,18,28)(6,13)(8,19)(9,21,17,27)(10,20)(11,15)(16,29,30,26)(22,25)(23,31,24,32)(33,35)(34,39)(36,44,49,45)(37,51,53,46)(38,47)(40,57,41,54)(42,60,50,55)(43,56)(48,61)(52,62)(58,64)(59,63);(1,4,6,15)(2,8,10,22)(3,11)(5,16)(7,19)(9,24)(12,26,28,29)(13,14)(17,23)(18,30)(20,25)(21,32,27,31)(33,36)(34,40)(35,44,47,45)(37,52)(38,49)(39,54,56,57)(41,43)(42,58)(46,61,51,62)(48,53)(50,59)(55,64,60,63);(4,15)(5,18)(7,20)(8,22)(16,30)(19,25)(26,29)(31,32)(33,36)(34,41)(35,45)(37,52)(38,49)(39,54)(40,43)(42,59)(44,47)(46,61)(48,53)(50,58)(51,62)(55,63)(56,57)(60,64);(1,5,6,18)(2,9)(3,12,13,28)(4,16)(7,21)(8,23,22,24)(10,17)(11,26)(14,29)(15,30)(19,31,25,32)(20,27)(33,37)(34,42,43,50)(35,46,47,51)(36,48,49,52)(38,53)(39,55)(40,58)(41,59)(44,61)(45,62)(54,63,57,64)(56,60)

Polycyclic presentation:

2^10|b^2=f,c^2=fgij,e^2=ghij,c^a=cfg,c^b=cgh,d^a=dfgj,d^b=dfgh,d^c=dfgij,e^a=ef,e^b=egi,e^c=efgh,e^d=ehij,f^c=fj,f^d=fj,f^e=fi,g^b=gi,g^c=gj,g^d=gj,g^e=gj,h^a=hi,h^b=hij,h^c=hij,h^e=hj

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284848
maximal normal subgroup of	#72284848

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