#135937

order	512
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1850
prime radical	C₂²×D₄	M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

341 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 341 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 24 points:

(5,8)(9,10,14,11,12,17,22,20,13,15,23,18,21,16,24,19);(1,2)(3,5)(4,6)(7,8)(9,11,14,17,12,20,22,15,13,18,23,16,21,19,24,10);(1,3)(2,5)(4,7)(6,8)(10,15)(11,18)(16,17)(19,20)

Polycyclic presentation:

2^9|a^2=f,b^2=f,d^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=ci,d^b=di,e^b=eg,d^c=dg

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, 43133);