#72287858

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

Polycyclic presentation:

2^10|a^2=hj,c^2=h,e^2=f,f^2=h,g^2=ij,i^2=j,b^a=bhj,c^a=cg,c^b=cj,d^a=dh,d^b=dhj,d^c=dh,e^a=egj,e^b=eh,e^c=eij,f^a=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, #72318841, #72319631
maximal normal subgroup of	#72318539, #72318841, #72319631

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