C₅×#10660 #11933380

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#10660

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂²×C₅	#399
prime radical	C₂²×C₅×D₄	Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ G

372 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 29
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 29 points:

(1,2,3,4,5)(6,7,8,11,9,12,13,10)(14,15,17,20,18,21,23,27)(16,22,24,28,25,29,26,19);(7,10)(8,13)(11,12)(14,16,23,24,18,25,17,26)(15,19,27,22,21,28,20,29);(7,10)(8,13)(11,12)(19,28)(22,29)

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 3648);