#151992

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

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

normal subgroups only

This group has 222 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=deg,d^2=gh,e^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ci,d^b=dg,e^b=eg,g^b=gh,h^b=hi

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, 59188);