#72318545

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
prime radical	#8904629	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×D₄ ◁ C₂²×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

5878 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)(9,12)(10,13)(11,14)(15,16)(20,31)(25,37)(27,41)(28,32)(30,44)(33,45)(36,39)(40,48);(1,3,5,10)(2,6,8,13)(4,9,11,15)(7,12,14,16)(17,18)(19,27,29,41)(20,25)(21,26)(22,24)(23,30,35,44)(28,42,32,46)(31,37)(33,40)(34,38)(36,43,39,47)(45,48);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,12,13,16)(17,19)(18,23)(20,32)(21,30,34,44)(22,29)(24,35)(25,39)(26,27,38,41)(28,31)(33,47,45,43)(36,37)(40,46,48,42);(2,8)(3,10)(4,11)(12,16)(18,24)(20,31)(23,35)(26,38)(27,41)(28,32)(33,45)(43,47);(2,8)(4,11)(6,13)(9,15)(17,20,22,31)(18,25,24,37)(19,28,29,32)(21,33,34,45)(23,36,35,39)(26,40,38,48)(27,42,41,46)(30,43,44,47)

Polycyclic presentation:

2^10|b^2=f,c^2=fhi,e^2=j,g^2=j,b^a=bi,c^a=ch,c^b=cg,d^a=dfi,d^b=dfij,d^c=dfi,e^a=efij,e^c=efij,e^d=ej,f^c=fj,g^a=gi,g^c=gij,h^b=hi,h^g=hj,i^c=ij

Conjugacy classes & character table

85 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318725, #72319813
maximal normal subgroup of	#72318725, #72319813

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