#491520

order	512
nilpotency class	4
isoclinism	isoclinic class of #491517
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	#3147	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

631 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 631 conjugacy classes of subgroups — too many to draw. Only its 174 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) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=fh,d^2=i,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^c=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, 398716);