C₂×#13671 #512845

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

Aliases

direct product

C₂×#13671

Subgroups

	subgroup	G/H
center, socle	C₂²	#2184
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	C₂×#3415	C₂
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

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

normal subgroups only

This group has 758 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);(3,4,10,17)(5,18,22,47)(6,23,29,56)(7,16,9,14)(8,32,34,48)(11,35,39,19)(12,30,41,20)(13,42,46,27)(15,31,50,25)(21,53,54,40)(24,49,59,66)(26,36,28,55)(33,43,63,65)(37,52,51,58)(38,62,64,57)(44,61,45,60);(3,5)(4,11)(6,24)(7,20)(8,33)(9,30)(10,22)(12,36)(13,43)(14,37)(15,49)(16,51)(17,39)(18,23)(19,27)(21,25)(26,57)(28,62)(29,59)(31,54)(32,38)(34,63)(35,42)(40,45)(41,55)(44,53)(46,65)(47,56)(48,64)(50,66)(52,61)(58,60);(3,5)(4,12)(6,19)(7,30)(8,21)(9,20)(10,22)(11,36)(13,38)(14,47)(15,40)(16,18)(17,41)(23,51)(24,27)(25,33)(26,58)(28,52)(29,35)(31,63)(32,43)(34,54)(37,56)(39,55)(42,59)(44,66)(45,49)(46,64)(48,65)(50,53)(57,60)(61,62)

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, 420041);