#135855

order	512
nilpotency class	3
isoclinism	isoclinic class of #1619
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1850
prime radical	#532	C₂×C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 281 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) = 256

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

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

Polycyclic presentation:

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

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, 43051);