C₂×#13668 #512839

order	512
nilpotency class	5
isoclinism	isoclinic class of #13667
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#13668

Subgroups

	subgroup	G/H
center, socle	C₂²	#2184
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	C₂×#3414	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 757 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) = 66
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 66 points:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#513025, C₂×#513024, C₂×#512843
maximal normal subgroup of	C₂×#513025, C₂×#513024, C₂×#512843

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420035);