C₂³×#499 #10575285

order	512
nilpotency class	3
isoclinism	isoclinic class of #465
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₂³×#499

Subgroups

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

7170 conjugacy classes of subgroups; only the 4258 normal subgroups are shown.

normal subgroups only

This group has 7170 conjugacy classes of subgroups — too many to draw. Only its 4258 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) = 512

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

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

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, 10482481);