#140387

order	512
nilpotency class	3
isoclinism	isoclinic class of #1685
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#121
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1939
prime radical	C₂³	#424
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

236 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 118 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,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,c^b=ci,d^b=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, 47583);