#72285477

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	#251911	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2253 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)(4,14)(5,9)(6,10)(8,17)(11,22)(12,20)(13,21)(15,19)(16,25)(18,26)(23,30)(24,29)(27,32)(28,31)(33,34)(35,40)(36,49)(37,42)(38,54)(39,44)(41,51)(43,55)(45,63)(46,57)(47,58)(48,59)(50,61)(52,56)(53,62)(60,64);(1,3,6,13)(2,7,10,21)(4,15,17,11)(5,12)(8,22,14,19)(9,24)(16,27)(18,29)(20,26)(23,32)(25,31)(28,30)(33,35,39,48)(34,40,44,59)(36,45,51,52)(37,46)(38,47)(41,56,49,63)(42,61)(43,58)(50,53)(54,60)(55,64)(57,62);(1,4)(2,8)(3,11)(5,16)(6,17)(7,19)(9,25)(10,14)(12,28)(13,15)(18,30)(20,31)(21,22)(23,26)(24,32)(27,29)(33,36,39,51)(34,41,44,49)(35,45,48,52)(37,47,53,64)(38,50,55,46)(40,56,59,63)(42,58,62,60)(43,57,54,61);(1,5,6,18)(2,9)(3,12)(4,16)(7,20,21,24)(8,23,14,25)(10,26)(11,27,15,28)(13,29)(17,30)(19,31)(22,32)(33,37)(34,42,44,62)(35,46,48,50)(36,47)(38,52,55,45)(39,53)(40,57)(41,60,49,58)(43,56)(51,64)(54,63)(59,61);(3,13)(5,18)(7,21)(8,14)(9,26)(19,22)(23,25)(31,32)(33,38)(34,43)(35,47)(36,50)(37,52)(39,55)(40,58)(41,61)(42,56)(44,54)(45,53)(46,51)(48,64)(49,57)(59,60)(62,63)

Polycyclic presentation:

2^10|b^2=fi,c^2=j,d^2=gij,b^a=bf,c^a=cfi,c^b=cij,d^a=dfi,d^b=dfg,d^c=dfgh,e^a=ehi,e^b=ehj,e^c=efghij,e^d=ehj,f^d=fi,f^e=fj,g^a=gi,g^b=gi,g^c=gij,g^e=gj,h^b=hj,h^c=hij,h^d=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, 2, F.6*F.9);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 4, F.7*F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.7*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.6*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);