C₂×#12414 #489143

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

Aliases

direct product

C₂×#12414

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

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

normal subgroups only

This group has 727 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) = 34
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 34 points:

(1,2);(3,4,9,13)(5,14,6,17)(7,12,8,11)(10,16,18,15)(20,25)(21,24)(22,29)(26,33)(27,34)(28,31)(30,32);(3,5,9,6)(4,10,13,18)(7,16,8,15)(11,17,12,14)(19,20,23,27)(21,26,24,32)(22,30,31,33)(25,28,34,29);(3,6,9,5)(4,10,13,18)(7,15,8,16)(11,17,12,14)(19,21,23,24)(20,26,27,32)(22,29,31,28)(25,33,34,30)

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, 396339);