C₂×#13726 #513044

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

Aliases

direct product

C₂×#13726

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

538 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 186 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)(9,19,24,45)(10,25,28,57)(11,29,31,59)(12,32,30,17)(14,33,36,21)(15,37,39,20)(16,40,41,26)(22,51,53,70)(23,54,52,44)(27,58,47,56)(34,64,65,69)(35,50,60,63)(38,48,55,66)(42,62,67,46)(43,61,68,49);(7,9,20,46,11,22,47,63,30,52,66,64,40,61,28,33,13,24,37,62,31,53,27,50,12,23,48,69,26,49,10,21)(8,14,25,43,16,34,55,54,32,60,58,70,59,67,39,45,18,36,57,68,41,65,38,44,17,35,56,51,29,42,15,19);(3,4,5,6)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420240);