C₂×#12510 #491447

order	512
nilpotency class	4
isoclinism	isoclinic class of #12503
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₈

Aliases

direct product

C₂×#12510

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

602 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 218 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) = 34
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 34 points:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398643);