#7623509

order	512
nilpotency class	4
isoclinism	isoclinic class of #33758
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
socle	C₂²	C₂³×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1320 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530705);