C₂×#13718 #512989

order	512
nilpotency class	5
isoclinism	isoclinic class of #13717
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#13718

Subgroups

	subgroup	G/H
center, socle	C₂²	#2233
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	C₂⁴	SD₃₂
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

361 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 361 conjugacy classes of subgroups — too many to draw. Only its 130 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)(5,15,16,44)(6,20,21,54)(7,26,29,62)(8,30,28,13)(10,32,33,17)(11,36,37,24)(12,40,42,23)(18,52,48,66)(19,53,31,47)(22,58,50,41)(25,45,57,56)(27,46,63,38)(34,59,65,61)(35,51,43,64)(39,60,55,49);(6,21)(11,37)(15,44)(17,32)(20,54)(22,50)(24,36)(25,57)(38,46)(39,55)(45,56)(47,53)(49,60)(51,64)(52,66)(59,61);(3,5,8,31,9,16,28,19)(4,10,13,43,14,33,30,35)(6,22,25,61,21,50,57,59)(7,27,23,18,29,63,40,48)(11,38,39,52,37,46,55,66)(12,41,26,34,42,58,62,65)(15,45,53,20,44,56,47,54)(17,49,64,24,32,60,51,36)

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#513025, C₂×#513024, C₂×#512993
maximal normal subgroup of	C₂×#513025, C₂×#513024, C₂×#512993

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420185);