#72284232

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

5904 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:

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

Polycyclic presentation:

2^10|c^2=f,d^2=g,e^2=j,c^a=cf,c^b=cij,d^a=dfi,d^b=di,d^c=dfg,e^a=eh,e^b=efgij,e^c=efghi,e^d=ehi,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

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284848
maximal normal subgroup of	#72284848

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