C₂×#13669 #512844

order	512
nilpotency class	5
isoclinism	isoclinic class of #13669
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#13669

Subgroups

	subgroup	G/H
center, socle	C₂²	#2184
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#504 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 768 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)(5,18)(6,23)(7,16)(8,32)(9,14)(10,17)(11,19)(12,20)(13,27)(15,25)(21,52)(22,45)(24,47)(26,35)(28,54)(29,55)(30,40)(31,48)(33,41)(34,46)(36,57)(37,56)(38,51)(39,53)(42,59)(43,60)(44,61)(49,50)(58,66)(62,64)(63,65);(3,5)(4,11)(6,24)(7,20)(8,33)(9,30)(10,22)(12,35)(13,41)(14,36)(15,47)(16,49)(17,38)(18,23)(19,27)(21,25)(26,56)(28,62)(29,58)(31,53)(32,37)(34,63)(39,43)(40,54)(42,52)(44,65)(45,55)(46,64)(48,66)(50,60)(51,61)(57,59);(3,5)(4,12)(6,19)(7,30)(8,21)(9,20)(10,22)(11,35)(13,37)(14,45)(15,39)(16,18)(17,40)(23,49)(24,27)(25,33)(26,57)(28,50)(29,51)(31,63)(32,41)(34,53)(36,55)(38,54)(42,66)(43,47)(44,64)(46,65)(48,52)(56,59)(58,61)(60,62)

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#512849
maximal normal subgroup of	C₂×#512849

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420040);