C₂×#13719 #513028

order	512
nilpotency class	5
isoclinism	isoclinic class of #13694
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₂×#13719

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	C₂×#2246	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

462 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 170 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) = 42
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 42 points:

(1,2);(3,4)(5,8)(6,7)(9,10)(12,13)(14,15)(16,26)(17,32)(19,23)(20,35)(21,34)(22,25)(24,37)(27,41)(28,31)(29,30)(33,40)(36,39)(38,42);(4,7)(8,10)(11,12,15,19,16,20,29,34,32,37,41,42,40,39,28,25,18,22,31,36,33,38,27,24,17,21,30,35,26,23,14,13);(3,5,6,9)(4,8,7,10)(11,13,14,23,26,35,30,21,17,24,27,38,33,36,31,22,18,25,28,39,40,42,41,37,32,34,29,20,16,19,15,12)

Conjugacy classes & character table

140 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, 420224);