#72285833

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

2845 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:

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

Polycyclic presentation:

2^10|c^2=fhj,d^2=ghij,e^2=gh,b^a=bf,c^a=cfij,c^b=cfh,d^a=dfg,d^b=dfij,d^c=df,e^a=ef,e^b=efhj,e^c=efgi,e^d=ei,f^c=fi,f^d=fij,f^e=fi,g^b=gij,g^c=gj,g^d=gj,g^e=gj,h^a=hi,h^c=hi,h^d=hj,h^e=hj

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340258, #72340284, #72340307, …
maximal normal subgroup of	#72340258, #72340284, #72340307, …

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