#140131

order	512
nilpotency class	3
isoclinism	isoclinic class of #1685
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₄

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	#1923
prime radical	C₂³	#435
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

235 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 235 conjugacy classes of subgroups — too many to draw. Only its 94 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)(3,5)(4,6)(7,8)(9,10,14,19,13,21,12,28,15,20,31,17,30,18,23,36)(11,22,26,37,25,16,29,32,27,34,39,40,38,33,24,35);(2,5)(6,7)(9,11,23,24,30,38,31,39,15,27,12,29,13,25,14,26)(10,16,28,40,18,22,19,32,20,33,36,37,21,34,17,35);(2,6)(4,8)(5,7)(9,12,30,14,15,23,13,31)(10,17,21,36,20,19,18,28)(11,24,38,39,27,29,25,26)(16,32,34,40,33,35,22,37)

Polycyclic presentation:

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

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, 47327);