#146923

order	512
nilpotency class	4
isoclinism	isoclinic class of #2218
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

502 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 210 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)(9,10,15,21,16,23,32,22)(11,24,26,36,27,18,28,40)(12,17,29,37,30,39,25,38)(13,31,33,19,34,20,14,35);(5,6,7,8)(9,11,14,25,15,26,13,12,16,27,33,29,32,28,34,30)(10,17,20,36,22,38,19,24,23,39,31,40,21,37,35,18);(2,4)(9,12,13,28,32,25,14,27,16,30,34,26,15,29,33,11)(10,18,19,38,21,40,20,17,23,24,35,37,22,36,31,39)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=hi,e^2=h,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=ghi,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, 54119);