C₂×#13690 #512912

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

Aliases

direct product

C₂×#13690

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

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

normal subgroups only

This group has 472 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);(4,6)(7,8)(9,19)(10,14)(11,27)(12,32)(13,34)(15,21)(16,30)(17,29)(18,26)(20,45)(22,52)(23,54)(24,46)(25,57)(28,43)(31,64)(33,41)(35,47)(36,48)(37,50)(38,53)(39,62)(40,61)(42,69)(44,51)(49,70)(55,68)(56,60)(58,67)(59,63)(65,66);(3,4,5,6)(8,14)(15,36)(16,37)(17,38)(18,34)(19,45)(27,46)(32,57)(39,66)(40,44)(41,67)(42,59)(43,68)(47,54)(52,70)(60,64);(7,9,10,20)(8,15,14,36)(11,28,24,55)(12,33,25,58)(13,35,26,23)(16,41,37,67)(17,43,38,68)(18,44,34,40)(19,46,45,27)(21,50,48,30)(22,53,49,29)(31,65,56,62)(32,39,57,66)(42,70,59,52)(47,60,54,64)(51,63,61,69)

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, 420108);