C₅×#9389 #11932109

order	1280
nilpotency class	3
isoclinism	isoclinic class of #9385
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₅×#9389

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#386
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₅×#392	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

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

normal subgroups only

This group has 496 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) = 37
transitive degree	μ_t(G) = 160

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

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

Conjugacy classes & character table

185 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 2377);