#72318565

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

5765 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)(3,9)(4,12)(5,8)(6,11)(7,13)(10,14)(15,16)(17,18,23,28)(19,24)(20,33,36,25)(21,26,22,27)(29,40)(30,42)(31,38)(32,41)(34,39)(35,44,37,43)(45,47)(46,48);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(19,29)(21,22)(24,38)(26,27)(30,45)(31,32)(34,46)(35,37)(39,47)(40,41)(42,48)(43,44);(1,4)(2,7)(3,10)(5,13)(6,14)(8,12)(9,15)(11,16)(17,19)(18,24,28,41)(20,34)(21,31,22,29)(23,32)(25,42,33,39)(26,40)(27,38)(30,36)(35,46,37,45)(43,48)(44,47);(2,8)(3,11)(4,13)(14,15)(17,20)(18,25)(19,30)(21,37)(22,35)(23,36)(24,39)(26,44)(27,43)(28,33)(29,45)(31,46)(32,34)(38,47)(40,48)(41,42);(2,8)(3,11)(7,12)(10,16)(17,21)(18,26)(19,31)(20,35)(22,23)(24,40)(25,43)(27,28)(29,32)(30,45)(33,44)(34,46)(36,37)(38,41)(39,47)(42,48)

Polycyclic presentation:

2^10|a^2=ij,c^2=h,g^2=j,b^a=bf,c^a=cfh,c^b=cg,d^a=dfij,d^b=dfi,d^c=dfij,e^a=ef,e^b=efj,e^c=eij,e^d=ej,f^c=fj,g^a=gi,g^b=gj,g^c=gij,g^e=gj,h^a=hj,h^b=hi,h^e=hj,h^g=hj,i^c=ij

Conjugacy classes & character table

79 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318725, #72319918
maximal normal subgroup of	#72318725, #72319918

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