C₂×#12387 #489213

order	512
nilpotency class	4
isoclinism	isoclinic class of #12316
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#12387

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×C₈×Q₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 730 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);(3,4,7,10)(5,11,12,18)(6,14,15,9)(8,16,17,13)(19,20,25,29)(21,30,33,49)(22,27,24,39)(23,35,38,47)(26,41,44,32)(28,34,46,37)(31,48,40,45)(36,42,50,43);(6,15)(9,14)(11,18)(13,16)(19,21,25,33)(20,26,29,44)(22,31,24,40)(23,36,38,50)(27,42,39,43)(28,45,46,48)(30,47,49,35)(32,34,41,37);(3,5,7,12)(4,8,10,17)(6,16,15,13)(9,18,14,11)(19,22,25,24)(20,27,29,39)(21,31,33,40)(23,34,38,37)(26,42,44,43)(28,35,46,47)(30,48,49,45)(32,36,41,50)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396409);