#495674

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator, socle	C₂³	C₂³×C₈
prime radical	#2388	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1009 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)(5,6)(7,8)(9,10,13,18,14,12,24,26,15,19,28,35,29,27,17,33)(11,16,22,32,23,25,31,39,20,30,36,40,37,21,34,38);(2,4)(9,11)(10,16)(12,25)(13,22)(14,23)(15,20)(17,34)(18,32)(19,30)(21,27)(24,31)(26,39)(28,36)(29,37)(33,38)(35,40);(6,8)(11,20)(16,30)(21,25)(22,36)(23,37)(31,34)(32,40)(38,39);(9,12,13,26,14,19,24,35,15,27,28,33,29,10,17,18)(11,21,22,38,23,16,31,32,20,25,36,39,37,30,34,40)

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402870);