C₂×#12514 #491452

order	512
nilpotency class	4
isoclinism	isoclinic class of #12506
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₈

Aliases

direct product

C₂×#12514

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

562 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 186 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) = 50
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 50 points:

(1,2);(3,4)(5,11)(6,14)(7,9)(8,12)(10,15)(13,17)(16,18)(19,20,27,34)(21,35,40,29)(22,41,36,28)(23,31,42,50)(24,30,44,48)(25,33,26,32)(37,47,43,46)(38,49,39,45);(3,5,6,12,7,13,15,18)(4,8,10,11,9,16,14,17)(19,21,24,37,25,38,42,22,27,40,44,43,26,39,23,36)(20,28,31,45,32,46,48,29,34,41,50,49,33,47,30,35);(4,9)(8,16)(10,14)(11,17)(19,22,42,39,26,37,24,40,27,36,23,38,25,43,44,21)(20,29,48,47,33,45,31,41,34,35,30,46,32,49,50,28)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398648);