#494219

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

1044 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)(9,10,14,19,15,11,20,23,13,18,27,33,28,22,16,29)(12,17,25,32,26,21,34,36,24,31,37,40,38,35,30,39);(2,4)(9,11,14,23,15,18,20,33,13,22,27,29,28,10,16,19)(12,21,25,36,26,31,34,40,24,35,37,39,38,17,30,32);(6,8)(9,12)(10,17)(11,21)(13,24)(14,25)(15,26)(16,30)(18,31)(19,32)(20,34)(22,35)(23,36)(27,37)(28,38)(29,39)(33,40);(12,24)(17,31)(21,35)(25,37)(26,38)(30,34)(32,40)(36,39)

Polycyclic presentation:

2^9|a^2=g,b^2=g,g^2=h,h^2=i,b^a=be,c^a=cf,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, 401415);