#140509

order	512
nilpotency class	3
isoclinism	isoclinic class of #1705
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#127
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1954
prime radical	C₂³	#408
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

356 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 134 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47705);