C₂×C₅×SD₁₂₈ #11936459

order	1280
nilpotency class	6
isoclinism	isoclinic class of D₆₄
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂×C₅×SD₁₂₈

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	D₃₂
commutator, Frattini, cosocle	C₃₂	C₂³×C₅
prime radical	C₂×C₅×D₃₂	C₂
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₈ ◁ C₂×C₅×C₁₆ ◁ C₂×C₅×C₃₂ ◁ G

134 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 71
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 3 generators on 71 points:

(1,2)(67,68,69,70,71);(3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66);(4,34)(5,65)(6,32)(7,63)(8,30)(9,61)(10,28)(11,59)(12,26)(13,57)(14,24)(15,55)(16,22)(17,53)(18,20)(19,51)(21,49)(23,47)(25,45)(27,43)(29,41)(31,39)(33,37)(36,66)(38,64)(40,62)(42,60)(44,58)(46,56)(48,54)(50,52)

Conjugacy classes & character table

350 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 6727);