#7625053

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624824
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

1193 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)(17,18,21,26)(19,28,31,43)(20,25,23,34)(22,37,36,48)(24,38,40,32)(27,35,41,29)(30,42,33,47)(39,45,44,46);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,15)(13,16)(17,18)(19,24)(20,34)(21,26)(22,27)(23,25)(28,32)(29,37)(30,44)(31,40)(33,39)(35,48)(36,41)(38,43)(42,45)(46,47);(17,19,29,45,23,33,36,38,21,31,35,46,20,30,22,32)(18,24,37,42,25,39,41,43,26,40,48,47,34,44,27,28);(3,10)(6,9)(11,16)(14,15)(17,20,21,23)(18,25,26,34)(19,30,31,33)(22,35,36,29)(24,39,40,44)(27,37,41,48)(28,42,43,47)(32,46,38,45);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)(18,26)(24,40)(25,34)(27,41)(28,43)(37,48)(39,44)(42,47)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532249);