#506316

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

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 ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

514 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 150 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=i,d^2=e,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,g^c=gh,h^c=hi

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, 413512);