#7624682

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624128
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 ◁ C₄×#41 ◁ C₈×#41 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1239 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)(19,28)(20,25)(21,33)(22,37)(23,27)(24,31)(26,35)(29,44)(30,46)(32,43)(34,40)(36,45)(38,42)(39,47)(41,48);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,15)(13,16)(19,29)(24,39)(28,44)(30,38)(31,47)(34,48)(40,41)(42,46);(17,19,23,29)(18,24,27,39)(20,30,22,38)(21,34,36,48)(25,40,37,41)(26,42,43,46)(28,45,44,33)(31,32,47,35);(3,10)(6,9)(11,16)(14,15)(17,20,23,22)(18,25,27,37)(19,30,29,38)(21,32,36,35)(24,40,39,41)(26,33,43,45)(28,46,44,42)(31,34,47,48);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)(17,20,23,22)(18,25,27,37)(19,30,29,38)(21,32,36,35)(24,40,39,41)(26,33,43,45)(28,46,44,42)(31,34,47,48)

Polycyclic presentation:

2^9|a^2=f,c^2=i,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,c^b=ci,d^b=df,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, 7531878);