C₂³×#510 #10585864

order	512
nilpotency class	4
isoclinism	isoclinic class of #509
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂³×#510

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂×D₈
commutator, Frattini, cosocle	C₈	C₂⁶
prime radical	C₂³×#135	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

8034 conjugacy classes of subgroups; only the 3490 normal subgroups are shown.

normal subgroups only

This group has 8034 conjugacy classes of subgroups — too many to draw. Only its 3490 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) = 38
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 38 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10493060);