C₂³×#488 #10575226

order	512
nilpotency class	3
isoclinism	isoclinic class of #465
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

direct product

C₂³×#488

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁶
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

8650 conjugacy classes of subgroups; only the 3746 normal subgroups are shown.

normal subgroups only

This group has 8650 conjugacy classes of subgroups — too many to draw. Only its 3746 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) = 30
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 30 points:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10482422);