#72287852

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	#7624085	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ #17330 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2201 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^10|a^2=h,c^2=hj,e^2=f,f^2=h,g^2=ij,i^2=j,b^a=bh,c^a=cg,c^b=cj,d^a=dhj,d^b=dh,d^c=dhj,e^a=ej,e^b=eh,e^c=eg,f^c=fj,g^a=gi,g^c=gi,g^e=gij,i^a=ij,i^c=ij,i^e=ij

Conjugacy classes & character table

109 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318539, #72318898, #72318999, …
maximal normal subgroup of	#72318539, #72318898, #72318999, …

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