C₂×#13688 #512911

order	512
nilpotency class	5
isoclinism	isoclinic class of #13661
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#13688

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

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

normal subgroups only

This group has 462 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) = 70
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 70 points:

(1,2);(4,6)(7,8,13,18)(9,19,23,50)(10,24,29,48)(11,30,33,64)(12,34,32,17)(14,36,39,20)(15,40,43,27)(16,44,46,26)(21,52,53,69)(22,54,35,49)(25,45,59,70)(28,62,61,56)(31,41,65,68)(37,63,66,58)(38,51,47,57)(42,60,55,67);(3,4,5,6);(7,9,13,23)(8,14,18,39)(10,25,29,59)(11,31,33,65)(12,35,32,22)(15,41,43,68)(16,45,46,70)(17,47,34,38)(19,48,50,24)(20,40,36,27)(21,44,53,26)(28,63,61,58)(30,37,64,66)(42,69,55,52)(49,56,54,62)(51,60,57,67)

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, 420107);