#512534

order	512
nilpotency class	4
isoclinism	isoclinic class of #511954
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

582 conjugacy classes of subgroups; only the 266 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 266 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) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 44 points:

(2,4)(5,6,8,10)(7,11,12,9)(13,14,16,19)(15,23,25,37)(17,30,28,43)(18,31,29,22)(20,34,32,26)(21,36,33,24)(27,42,41,39)(35,40,38,44);(5,7,8,12)(6,9,10,11)(14,19)(20,32)(21,33)(22,31)(23,37)(30,43)(35,38)(39,42);(13,15,24,40,18,27,28,34,16,25,36,44,29,41,17,26)(14,20,30,39,22,35,33,37,19,32,43,42,31,38,21,23);(1,2,3,4)(15,25)(20,32)(23,37)(26,34)(27,41)(35,38)(39,42)(40,44)

Polycyclic presentation:

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

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, 419730);