#7622785

order	512
nilpotency class	2
isoclinism	isoclinic class of #3484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂×C₈	C₂⁵
commutator, socle	C₂²	C₂⁴×C₈
prime radical	C₂×2₋¹⁺⁴	C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1314 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,22,27,23,24,30,31,21,20,26,29,19,25,28,32);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,21,23)(18,24,20,25)(22,28,26,30)(27,31,29,32);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,19,21,23)(18,25,20,24)(22,28,26,30)(27,32,29,31);(2,8)(6,9)(7,16)(14,15);(17,20,22,29,23,25,30,32,21,18,26,27,19,24,28,31)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,e^2=g,g^2=h,h^2=i,b^a=bf,d^a=dfi,c^b=cfi,e^b=ei

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529981);