C₂×#12412 #489144

order	512
nilpotency class	4
isoclinism	isoclinic class of #12324
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#12412

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

711 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 711 conjugacy classes of subgroups — too many to draw. Only its 162 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);(3,4,9,13)(5,14,6,17)(7,12,8,11)(10,16,18,15)(19,20,25,29)(21,30,33,48)(22,27,24,39)(23,35,38,46)(26,41,43,32)(28,34,47,37)(31,49,40,45)(36,42,50,44);(3,5,9,6)(4,10,13,18)(7,16,8,15)(11,17,12,14)(19,21,25,33)(20,26,29,43)(22,31,24,40)(23,36,38,50)(27,42,39,44)(28,45,47,49)(30,46,48,35)(32,34,41,37);(3,6,9,5)(4,10,13,18)(7,15,8,16)(11,17,12,14)(19,22,25,24)(20,27,29,39)(21,31,33,40)(23,37,38,34)(26,42,43,44)(28,46,47,35)(30,45,48,49)(32,50,41,36)

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396340);