C₅×#11960 #11934680

order	1280
nilpotency class	4
isoclinism	isoclinic class of #11858
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#11960

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₄×C₈	C₂³×C₅
prime radical	C₅×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

392 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 118 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)(14,15,19,25,20,23,35,41,21,26,33,43,17,32,18,24)(16,27,30,40,28,42,29,37,31,44,34,22,36,38,45,39);(7,10)(8,12)(14,16)(15,22)(17,28)(18,34)(19,29)(20,36)(21,31)(23,37)(24,42)(25,27)(26,40)(30,35)(32,39)(33,45)(38,41)(43,44);(7,10)(11,13)(14,17,21,20)(15,23,26,32)(16,28,31,36)(18,33,35,19)(22,37,40,39)(24,25,41,43)(27,38,44,42)(29,34,45,30)

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, 4948);