C₂×#13698 #512936

order	512
nilpotency class	5
isoclinism	isoclinic class of #13697
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#13698

Subgroups

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

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

normal subgroups only

This group has 436 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) = 70
transitive degree	μ_t(G) = 256

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

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

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, 420132);