#506313

order	512
nilpotency class	4
isoclinism	isoclinic class of #500829
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

524 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 154 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);(6,9)(10,11)(13,14,17,21,29,36,43,39,18,22,31,38,16,20,28,35)(15,23,25,32,24,40,42,34,27,41,44,33,26,37,30,19);(2,4)(13,15,18,27)(14,19,22,34)(16,24,29,26)(17,30,31,42)(20,32,36,33)(21,37,38,40)(23,39,41,35)(25,43,44,28);(5,7,8,12)(6,10,9,11)(13,16,18,29)(14,20,22,36)(15,24,27,26)(17,28,31,43)(19,32,34,33)(21,35,38,39)(23,40,41,37)(25,42,44,30)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413509);