C₂×#12389 #489235

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

Aliases

direct product

C₂×#12389

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

721 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 721 conjugacy classes of subgroups — too many to draw. Only its 154 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)(19,20,24,28)(21,29,31,47)(22,33,35,49)(23,38,36,27)(25,40,41,30)(26,43,45,37)(32,48,39,44)(34,46,50,42);(6,15)(9,14)(11,18)(13,16)(19,21)(20,25)(22,34)(23,39)(24,31)(26,44)(27,46)(28,41)(29,33)(30,37)(32,36)(35,50)(38,42)(40,43)(45,48)(47,49);(3,5,7,12)(4,8,10,17)(6,16,15,13)(9,18,14,11)(22,35)(26,45)(29,47)(30,40)(33,49)(34,50)(37,43)(44,48)

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, 396431);