#494405

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator, socle	C₂³	C₂³×C₈
prime radical	C₂×#532	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1358 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)(9,10,14,19,15,20,29,36,13,18,27,34,28,35,30,40)(11,16,22,12,23,25,38,26,21,31,37,24,17,32,33,39);(1,2,3,4)(9,11,14,22,15,23,29,38,13,21,27,37,28,17,30,33)(10,16,19,12,20,25,36,26,18,31,34,24,35,32,40,39);(6,8)(9,12)(10,17)(11,20)(13,24)(14,25)(15,26)(16,30)(18,23)(19,33)(21,35)(22,36)(27,32)(28,39)(29,31)(34,38)(37,40);(10,18)(12,24)(16,31)(19,34)(20,35)(25,32)(26,39)(36,40)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,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, 401601);