#146330

order	512
nilpotency class	4
isoclinism	isoclinic class of #509
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₈	C₄³
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₂×#107
prime radical	C₂⁴	C₂×Q₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

650 conjugacy classes of subgroups; only the 322 normal subgroups are shown.

normal subgroups only

This group has 650 conjugacy classes of subgroups — too many to draw. Only its 322 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) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=hi,e^2=i,f^2=h,h^2=i,b^a=bd,d^a=dh,f^a=fhi,h^a=hi,c^b=ci

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53526);