C₅×#12398 #11935118

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12316
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#12398

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#447
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₅
prime radical	C₂×C₅×D₄	D₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×C₈×D₄ ◁ G

286 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 53
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 3 generators on 53 points:

(1,2,3,4,5)(6,7,11,15)(8,16,19,20)(9,14,10,13)(12,18,21,17)(22,23,27,31)(24,32,35,51)(25,36,40,50)(26,41,38,30)(28,43,45,33)(29,46,48,39)(34,52,42,47)(37,49,53,44);(6,8,11,19)(7,12,15,21)(9,18,10,17)(13,20,14,16)(22,24,27,35)(23,28,31,45)(25,37,40,53)(26,42,38,34)(29,47,48,52)(30,49,41,44)(32,50,51,36)(33,46,43,39);(7,13)(9,10)(12,20)(14,15)(16,21)(17,18)

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, 5386);