C₅×#12476 #11935196

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12430
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#12476

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#466
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₅
prime radical	C₅×#493	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×C₈×Q₈ ◁ G

214 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 37
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 3 generators on 37 points:

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

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, 5464);