#493206

order	512
nilpotency class	2
isoclinism	isoclinic class of #534
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator	C₂²	C₂²×C₄×C₈
socle	C₂³	C₂³×C₈
prime radical	C₂²×D₄	C₂×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

800 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=g,d^2=f,f^2=h,h^2=i,b^a=be,c^a=ci,d^a=di,c^b=ci

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, 400402);