C₂×#13716 #512976

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

Aliases

direct product

C₂×#13716

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

378 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 74
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 74 points:

(1,2);(3,4)(5,8)(6,7)(9,10)(11,12,17,23)(13,18,27,40)(14,19,31,43)(15,32,34,61)(16,35,33,22)(20,44,46,28)(21,47,48,29)(24,36,53,64)(25,54,56,74)(26,57,55,39)(30,49,58,42)(37,65,67,50)(38,68,69,51)(41,60,45,59)(52,70,71,63)(62,73,66,72);(4,7)(8,10)(11,13,17,27)(12,18,23,40)(14,24,31,53)(15,25,34,56)(16,26,33,55)(19,36,43,64)(20,37,46,67)(21,38,48,69)(22,39,35,57)(28,50,44,65)(29,51,47,68)(30,52,58,71)(32,54,61,74)(41,62,45,66)(42,63,49,70)(59,72,60,73);(3,5,6,9)(4,8,7,10)(12,19)(14,28)(15,29)(16,33)(18,36)(20,21)(22,49)(23,43)(24,50)(25,51)(26,55)(30,60)(31,44)(32,41)(34,47)(35,42)(37,38)(39,70)(40,64)(45,61)(46,48)(52,73)(53,65)(54,62)(56,68)(57,63)(58,59)(66,74)(67,69)(71,72)

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, 420172);