C₂×#13709 #512969

order	512
nilpotency class	5
isoclinism	isoclinic class of #13709
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#13709

Subgroups

	subgroup	G/H
center, socle	C₂²	#2226
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	C₂×#508	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 342 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) = 66
transitive degree	μ_t(G) = 128

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

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#513026, C₂×#513024, C₂×#512972
maximal normal subgroup of	C₂×#513026, C₂×#513024, C₂×#512972

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420165);