#512356

order	512
nilpotency class	4
isoclinism	isoclinic class of #512000
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#476
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

494 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 156 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 — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=efi,d^2=e,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gh,h^a=hi,c^b=ci,g^d=gh,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419552);