#7624729

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

1228 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)(17,18,24,29)(19,25,20,32)(21,27,23,40)(22,36,39,48)(26,31,45,34)(28,35,46,38)(30,41,33,43)(37,44,47,42);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,15)(13,16)(17,19,24,20)(18,25,29,32)(21,33,23,30)(22,31,39,34)(26,36,45,48)(27,43,40,41)(28,42,46,44)(35,37,38,47);(17,20,24,19)(18,26,29,45)(21,33,23,30)(22,37,39,47)(25,36,32,48)(27,44,40,42)(28,41,46,43)(31,38,34,35);(3,10)(6,9)(11,16)(14,15);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)(17,21,24,23)(18,27,29,40)(19,30,20,33)(22,35,39,38)(25,41,32,43)(26,44,45,42)(28,36,46,48)(31,47,34,37)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^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,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, 7531925);