#506425

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

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

normal subgroups only

This group has 522 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=eg,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, 413621);