C₅×#12021 #11934741

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12019
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₅×#12021

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#106
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₄×C₈	C₂³×C₅
prime radical	C₂³×C₅	#106
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

216 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 45
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 3 generators on 45 points:

(1,2,3,4,5)(6,7,8,11,9,10,12,13)(14,15)(16,25,28,40)(17,22)(18,31)(19,34)(20,24)(21,27,36,43)(23,32)(26,39,29,41)(30,38)(33,45)(35,42,37,44);(14,16,18,27,19,26,32,42,20,28,33,43,17,29,30,44)(15,21,23,25,22,37,31,41,24,36,38,40,34,35,45,39);(7,10)(11,13)(14,17,20,19)(15,22,24,34)(16,26,28,29)(18,30,33,32)(21,35,36,37)(23,31,38,45)(25,39,40,41)(27,42,43,44)

Conjugacy classes & character table

260 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5009);