C₅×#13686 #11936406

order	1280
nilpotency class	5
isoclinism	isoclinic class of #13661
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#13686

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₅
prime radical	C₅×#2246	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₄×C₅×C₈ ◁ C₄×C₅×C₁₆ ◁ G

220 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 41
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 3 generators on 41 points:

(1,2,3,4,5)(7,9)(11,16)(12,20)(13,22)(14,26)(17,32)(18,35)(19,31)(21,25)(23,39)(24,29)(27,33)(28,30)(34,40)(36,37)(38,41);(6,7,8,9);(10,11,15,19)(12,21,24,38)(13,25,27,41)(14,28,26,18)(16,29,31,20)(17,33,34,22)(23,40,36,32)(30,37,35,39)

Conjugacy classes & character table

245 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 6674);