#151960

order	512
nilpotency class	5
isoclinism	isoclinic class of #2263
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₁₆
commutator	C₁₆	C₂×C₄²
Frattini, cosocle	C₂²×C₁₆	C₂³
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

300 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=ei,d^2=gh,f^2=g,g^2=hi,h^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gh,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, 59156);