C₂×#12397 #489076

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

Aliases

direct product

C₂×#12397

Subgroups

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

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

normal subgroups only

This group has 704 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);(3,4,8,12)(5,13,16,17)(6,11,7,10)(9,15,18,14)(19,20,25,29)(21,30,33,48)(22,27,24,39)(23,35,38,49)(26,41,44,32)(28,34,46,37)(31,47,40,45)(36,42,50,43);(3,5,8,16)(4,9,12,18)(6,15,7,14)(10,17,11,13)(19,21)(20,26)(22,31)(23,36)(24,40)(25,33)(27,42)(28,45)(29,44)(30,35)(32,37)(34,41)(38,50)(39,43)(46,47)(48,49);(4,10)(6,7)(9,17)(11,12)(13,18)(14,15)(19,22,25,24)(20,27,29,39)(21,31,33,40)(23,34,38,37)(26,42,44,43)(28,35,46,49)(30,47,48,45)(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, 396272);