C₂×#13697 #512935

order	512
nilpotency class	5
isoclinism	isoclinic class of #13697
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₂×#13697

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
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

488 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 146 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)(9,14)(10,18)(11,21)(15,23)(16,32)(17,33)(19,35)(20,29)(22,34)(24,25)(26,31)(27,28)(30,38)(36,37);(7,8,14,25,10,16,27,33,21,31,35,38,34,36,20,23,12,15,29,37,22,30,19,26,11,17,28,32,18,24,9,13);(4,6)(8,15)(13,23)(16,30)(17,31)(24,36)(25,37)(26,33)(32,38)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#513082
maximal normal subgroup of	C₂×#513082

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420131);