#146922

order	512
nilpotency class	4
isoclinism	isoclinic class of #2215
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

522 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 218 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)(9,10,15,21,16,23,29,22)(11,18,25,36,26,37,27,32)(12,17,24,33,28,35,40,34)(13,20,14,19,30,38,31,39);(5,6,7,8)(9,11,14,24,15,25,30,28,16,26,31,40,29,27,13,12)(10,17,20,32,22,34,39,37,23,35,38,36,21,33,19,18);(2,4)(9,12,13,27,29,40,31,26,16,28,30,25,15,24,14,11)(10,18,19,33,21,36,38,35,23,37,39,34,22,32,20,17)

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

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, 54118);