#491514

order	512
nilpotency class	4
isoclinism	isoclinic class of #491511
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#19661	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

649 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 649 conjugacy classes of subgroups — too many to draw. Only its 166 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=fhi,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,h^a=hi,c^b=cg,d^b=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398710);