#496163

order	512
nilpotency class	4
isoclinism	isoclinic class of #2218
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	C₂³	C₂×#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

484 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 234 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=fh,d^2=eg,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403359);