C₂×#12512 #491450

order	512
nilpotency class	4
isoclinism	isoclinic class of #12506
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₈

Aliases

direct product

C₂×#12512

Subgroups

	subgroup	G/H
center, socle	C₂³	#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

620 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 186 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)(5,11)(6,14)(7,9)(8,12)(10,15)(13,17)(16,18)(19,20)(21,29)(22,32)(23,31)(24,33)(25,30)(26,28)(27,34);(3,5,6,12,7,13,15,18)(4,8,10,11,9,16,14,17)(19,21,29,20,25,31,24,27,26,22,32,28,34,33,23,30);(4,9)(8,16)(10,14)(11,17)(19,22,32,20,25,33,23,27,26,21,29,28,34,31,24,30)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398646);