#7624672

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
prime radical	C₂×#41	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₈×#41 ◁ 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)(17,18)(19,24)(20,30)(21,34)(22,38)(23,27)(25,32)(26,36)(28,44)(29,43)(31,42)(33,47)(35,39)(37,46)(40,48)(41,45);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,15)(13,16)(17,19,23,22)(18,24,27,38)(20,31,33,28)(21,29,37,36)(25,40,41,39)(26,34,43,46)(30,42,47,44)(32,48,45,35);(17,20,23,33)(18,25,27,41)(19,28,22,31)(21,35,37,48)(24,39,38,40)(26,42,43,44)(29,45,36,32)(30,46,47,34);(3,10)(6,9)(11,16)(14,15);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)

Polycyclic presentation:

2^9|a^2=f,b^2=i,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,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, 7531868);