#136810

order	512
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

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	#1869
prime radical	#2388	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 549 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) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 44006);