#7622648

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×#41	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1326 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=fg,g^2=h,h^2=i,b^a=bf,c^a=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, 7529844);