#7622671

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	#3475	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1376 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=f,g^2=h,h^2=i,b^a=bf,c^a=ci,c^b=ci,e^d=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, 7529867);