#7624742

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

1199 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)(19,29)(20,22)(21,33)(24,37)(25,28)(27,42)(30,46)(31,45)(32,35)(34,36)(38,48)(39,47)(40,41)(43,44);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,15)(13,16)(17,18)(19,24)(20,25)(21,27)(22,28)(23,26)(29,37)(30,38)(31,39)(32,40)(33,42)(34,43)(35,41)(36,44)(45,47)(46,48);(17,19,23,31)(18,24,26,39)(20,30,22,36)(21,34,35,46)(25,38,28,44)(27,43,41,48)(29,32,45,33)(37,40,47,42);(3,10)(6,9)(11,16)(14,15)(17,20,23,22)(18,25,26,28)(19,30,31,36)(21,32,35,33)(24,38,39,44)(27,40,41,42)(29,34,45,46)(37,43,47,48);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)(18,26)(24,39)(25,28)(27,41)(37,47)(38,44)(40,42)(43,48)

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=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, 7531938);