#505690

order	512
nilpotency class	4
isoclinism	isoclinic class of #500834
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₄	D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

561 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 561 conjugacy classes of subgroups — too many to draw. Only its 162 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,d^2=eh,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 412886);