C₅×#8556 #11931276

order	1280
nilpotency class	3
isoclinism	isoclinic class of #8541
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#8556

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#379
commutator	C₂⁴	C₂²×C₄×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
prime radical	C₂⁵×C₅	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₄×C₅ ◁ G

842 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 842 conjugacy classes of subgroups — too many to draw. Only its 150 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)(6,7,10,17,11,15,9,22)(8,18,19,32,20,13,25,29)(12,16,26,31,23,30,24,14)(21,34,36,33,35,28,37,27);(6,8,12,21)(7,13,16,28)(9,19,24,36)(10,25,26,37)(11,20,23,35)(14,27,22,29)(15,18,30,34)(17,32,31,33);(6,9,23,26)(7,14,30,17)(8,19,35,37)(10,12,24,11)(13,27,34,32)(15,31,16,22)(18,33,28,29)(20,25,21,36)

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, 1544);