C₂×#13693 #512927

order	512
nilpotency class	5
isoclinism	isoclinic class of #13691
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#13693

Subgroups

	subgroup	G/H
center, socle	C₂²	#2206
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

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

normal subgroups only

This group has 454 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);(3,4,11,18)(5,16,22,37)(6,23,26,55)(7,17,10,14)(8,28,32,60)(9,12,21,38)(13,39,41,25)(15,43,46,30)(19,49,50,66)(20,42,52,45)(24,56,34,54)(27,36,31,59)(29,47,61,64)(33,44,53,65)(35,62,63,58)(40,57,48,51);(3,5,10,21,11,22,7,9)(4,12,17,37,18,38,14,16)(6,19,34,53,26,50,24,33)(8,20,31,43,32,52,27,30)(13,35,48,64,41,63,40,47)(15,36,45,60,46,59,42,28)(23,44,56,66,55,65,54,49)(25,29,51,62,39,61,57,58);(3,6)(4,13)(5,19)(7,24)(8,29)(9,33)(10,34)(11,26)(12,35)(14,40)(15,44)(16,47)(17,48)(18,41)(20,51)(21,53)(22,50)(23,28)(25,30)(27,58)(31,62)(32,61)(36,56)(37,64)(38,63)(39,43)(42,49)(45,66)(46,65)(52,57)(54,59)(55,60)

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420123);