C₂×#13722 #513032

order	512
nilpotency class	5
isoclinism	isoclinic class of #13697
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₂×#13722

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	C₂²×D₁₆	C₂²
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

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

normal subgroups only

This group has 416 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) = 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)(11,12)(13,25)(14,21)(15,20)(16,36)(17,38)(18,24)(19,23)(22,30)(26,39)(27,33)(28,35)(29,34)(31,41)(32,40)(37,42);(4,7)(8,10)(11,12,15,21,16,22,33,29,19,38,25,41,39,42,32,28,18,24,35,40,37,26,31,13,17,23,34,27,30,36,14,20);(3,5,6,9)(4,8,7,10)(11,13,14,26,30,40,34,24,17,28,31,42,37,41,35,38,18,29,32,22,39,21,25,12,19,20,33,36,16,27,15,23)

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, 420228);