#7624665

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624118
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

1249 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)(19,24)(20,30)(21,33)(22,37)(23,27)(25,31)(26,35)(28,44)(29,43)(32,47)(34,39)(36,46)(38,42)(40,48)(41,45);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,15)(13,16)(17,19,23,22)(18,24,27,37)(20,28,32,38)(21,29,36,35)(25,39,41,40)(26,33,43,46)(30,44,47,42)(31,34,45,48);(17,20,23,32)(18,25,27,41)(19,28,22,38)(21,34,36,48)(24,39,37,40)(26,42,43,44)(29,45,35,31)(30,46,47,33);(3,10)(6,9)(11,16)(14,15)(17,19,23,22)(18,24,27,37)(20,28,32,38)(21,29,36,35)(25,39,41,40)(26,33,43,46)(30,44,47,42)(31,34,45,48);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)(17,19,23,22)(18,24,27,37)(20,28,32,38)(21,29,36,35)(25,39,41,40)(26,33,43,46)(30,44,47,42)(31,34,45,48)

Polycyclic presentation:

2^9|a^2=f,b^2=i,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,d^b=df,g^c=gh,h^c=hi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531861);