#7624788

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624184
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #529 ◁ #2966 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1202 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=i,e^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gh,h^a=hi,d^b=dfi,e^b=ei,g^c=gh,h^c=hi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531984);