#496090

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator, socle	C₂³	C₂³×C₈
prime radical	#2397	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

810 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=f,d^2=g,g^2=h,h^2=i,b^a=be,c^a=cf,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, 403286);