#494220

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	C₂²×#41	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1040 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)(9,10,14,19,15,20,30,36,13,18,28,34,29,35,32,40)(11,21,23,25,24,37,38,39,22,16,31,12,17,26,33,27);(2,4)(9,11,14,23,15,24,30,38,13,22,28,31,29,17,32,33)(10,16,19,12,20,26,36,27,18,21,34,25,35,37,40,39);(6,8)(9,12)(10,17)(11,20)(13,25)(14,26)(15,27)(16,32)(18,24)(19,33)(21,30)(22,35)(23,36)(28,37)(29,39)(31,40)(34,38);(10,18)(12,25)(16,21)(19,34)(20,35)(26,37)(27,39)(36,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=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, 401416);