#491513

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	#33717	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)(11,19)(12,23)(13,24)(15,26)(17,32)(18,31)(20,25)(21,36)(22,35)(27,30)(28,37)(29,40)(33,34)(38,39);(1,2,3,5,4,6,7,8)(9,10)(11,15)(12,17)(13,18)(14,16)(19,26)(20,27)(21,29)(22,28)(23,32)(24,31)(25,30)(33,38)(34,39)(35,37)(36,40);(2,6)(5,8)(9,11,20,36,13,21,12,19,14,22,23,34,24,33,25,35)(10,15,27,40,18,29,17,26,16,28,32,39,31,38,30,37);(10,16)(15,28)(17,30)(18,31)(26,37)(27,32)(29,38)(39,40)

Polycyclic presentation:

2^9|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, 398709);