#72285949

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

2358 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)(5,17)(6,10)(8,16)(9,18)(11,25)(12,23)(13,21)(15,22)(19,27)(20,29)(24,30)(26,31)(28,32)(33,34)(35,39)(36,40)(37,48)(38,42)(41,49)(43,50)(44,51)(45,52)(46,55)(47,54)(53,56)(57,61)(58,64)(59,63)(60,62);(1,3,6,13)(2,7,10,21)(4,11,16,27)(5,12)(8,19,14,25)(9,23)(15,28)(17,20)(18,29)(22,31)(24,32)(26,30)(33,35)(34,39)(36,44,47,56)(37,45,49,57)(38,43)(40,53,54,51)(41,52,48,61)(42,50)(46,59)(55,63)(58,60)(62,64);(1,4)(2,8)(3,11)(5,15)(6,16)(7,19)(9,24)(10,14)(12,28)(13,27)(17,22)(18,30)(20,31)(21,25)(23,32)(26,29)(35,43)(39,50)(40,54)(41,48)(44,56)(45,57)(58,59)(63,64);(1,5)(2,9,10,17)(3,12,13,29)(4,15,16,30)(6,18)(7,20)(8,22)(11,26)(14,24)(19,31,25,32)(21,23)(27,28)(33,36,38,47)(34,40)(35,44)(37,46)(39,51,50,53)(41,55,48,62)(42,54)(43,56)(45,59,57,58)(49,60)(52,64)(61,63);(3,13)(5,18)(7,21)(8,14)(9,17)(19,25)(22,24)(31,32)(33,37,38,49)(34,41,42,48)(35,45,43,57)(36,46,47,60)(39,52,50,61)(40,55,54,62)(44,58,56,59)(51,63,53,64)

Polycyclic presentation:

2^10|b^2=fij,d^2=gj,e^2=j,b^a=bf,c^a=cfij,c^b=cj,d^a=dfj,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

55 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286044
maximal normal subgroup of	#72286044

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