#7624717

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624150
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

1200 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)(18,24)(19,28)(20,23)(22,34)(25,39)(27,44)(29,47)(30,46)(32,33)(35,36)(37,41)(38,43)(40,48)(42,45);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,15)(13,16)(17,18)(19,25)(20,31)(21,24)(22,27)(23,26)(28,41)(29,42)(30,37)(32,44)(33,38)(34,43)(35,45)(36,40)(39,46)(47,48);(17,19,21,30)(18,25,24,37)(20,29,23,36)(22,35,33,47)(26,40,31,42)(27,45,38,48)(28,32,46,34)(39,43,41,44);(3,10)(6,9)(11,16)(14,15)(18,24)(25,37)(26,31)(27,38)(39,41)(40,42)(43,44)(45,48);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)(17,20,21,23)(18,26,24,31)(19,29,30,36)(22,32,33,34)(25,40,37,42)(27,43,38,44)(28,35,46,47)(39,45,41,48)

Polycyclic presentation:

2^9|a^2=fi,c^2=i,e^2=f,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,g^c=gh,h^c=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531913);