#72310953

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

2956 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 48 points:

(6,9)(7,14)(11,16)(13,15)(17,18)(19,27)(20,24)(21,25)(22,26)(23,30)(28,41)(29,42)(31,37)(32,46)(33,40)(34,36)(35,39)(38,44)(43,47)(45,48);(1,2)(3,9,10,6)(4,7,12,14)(5,8)(11,13)(15,16)(17,18)(19,23,30,27)(20,24)(21,34)(22,26)(25,36)(28,37,31,41)(29,42,44,38)(32,39)(33,40)(35,46)(43,47,45,48);(1,3)(2,6)(4,11,12,16)(5,10)(7,15,14,13)(8,9)(17,19)(18,23,26,27)(20,31)(21,29)(22,30)(24,37,40,41)(25,38,34,42)(28,33)(32,43)(35,45)(36,44)(39,48,46,47);(3,10)(6,9)(17,20)(18,24)(19,28)(21,35)(22,33)(23,37)(25,39)(26,40)(27,41)(29,43)(30,31)(32,36)(34,46)(38,48)(42,47)(44,45);(1,4,5,12)(2,7,8,14)(3,11,10,16)(6,13,9,15)(17,21,22,36)(18,25,26,34)(19,29,30,44)(20,32,33,35)(23,38,27,42)(24,39,40,46)(28,43,31,45)(37,47,41,48)

Polycyclic presentation:

2^10|b^2=f,c^2=gh,e^2=i,b^a=bf,c^a=cfh,c^b=cfg,d^c=dghi,e^a=eh,e^b=ehi,e^c=ehj,e^d=efghij,f^c=fi,f^e=fj,g^a=gij,g^b=gi,g^c=gj,g^d=gij,g^e=gj,h^b=hj,h^c=hj,h^d=hij

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310969, #72340284, #72340311, …
maximal normal subgroup of	#72310969, #72340284, #72340311, …

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