#72318529

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂⁵
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×D₄ ◁ C₂²×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

5835 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:

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

Polycyclic presentation:

2^10|b^2=f,c^2=h,g^2=j,b^a=bi,c^a=ch,c^b=cfgj,d^a=dfi,d^b=dfij,d^c=dfi,e^a=efj,e^b=ef,e^c=ei,e^d=ej,f^c=fj,g^a=gi,g^c=gi,g^e=gj,h^b=hi,h^e=hj,h^g=hj,i^c=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318725
maximal normal subgroup of	#72318725

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