C₂×#13706 #512958

order	512
nilpotency class	5
isoclinism	isoclinic class of #13661
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#13706

Subgroups

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

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

normal subgroups only

This group has 366 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) = 512

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

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420154);