#72285129

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#35346
commutator, Frattini, cosocle	C₂⁵	C₂⁵
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

5676 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|b^2=i,c^2=fi,d^2=gij,b^a=bi,c^a=cf,d^a=df,d^b=di,d^c=dfg,e^a=eh,e^b=efgi,e^c=efghij,e^d=ehij,f^d=fi,f^e=fj,g^a=gi,g^c=gi,g^e=gj,h^b=hi,h^c=hij,h^d=hj

Conjugacy classes & character table

85 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.9);;
gap> SetPower(coll, 3, F.6*F.9);;
gap> SetPower(coll, 4, F.7*F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.7*F.9);;
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.9*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, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
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);