C₂×#12511 #491448

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

Aliases

direct product

C₂×#12511

Subgroups

	subgroup	G/H
center, socle	C₂³	#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 596 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) = 256

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

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, 398644);