#124241

order	512
nilpotency class	4
isoclinism	isoclinic class of #1366
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#333
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#333
prime radical	C₂³	C₂×#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

308 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=fgh,d^2=gi,e^2=h,f^2=i,h^2=i,b^a=bd,d^a=df,f^a=fi,g^a=gi,d^b=dg,e^b=efg,f^b=fi,h^b=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31437);