C₂×#13696 #512933

order	512
nilpotency class	5
isoclinism	isoclinic class of #13694
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#13696

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	C₂×#2215	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 472 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) = 38
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 38 points:

(1,2);(3,4)(5,6)(8,13)(9,18)(10,19)(11,22)(14,21)(15,30)(16,33)(17,27)(20,29)(23,36)(24,25)(26,31)(28,37)(32,38)(34,35);(7,8,14,28,10,15,20,26,22,31,35,30,36,37,9,13,12,17,18,25,23,32,34,33,11,16,29,38,19,24,21,27);(4,6)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420129);