#495792

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator, socle	C₂³	C₂³×C₈
prime radical	#392	C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

721 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)(5,6)(7,8)(9,10,14,18,15,19,27,32,13,12,17,24,26,25,31,37)(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)(13,20)(14,21)(15,22)(17,30)(18,28)(19,29)(24,35)(25,36)(26,33)(27,34)(31,39)(32,38)(37,40);(6,8)(10,12)(11,20)(18,24)(19,25)(21,30)(22,33)(32,37)(34,39);(9,12,14,24,15,25,27,37,13,10,17,18,26,19,31,32)(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,d^2=g,g^2=h,h^2=i,b^a=be,c^a=cf,c^b=ci,d^b=di,d^c=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, 402988);