C₅×#11524 #11934244

order	1280
nilpotency class	4
isoclinism	isoclinic class of #11521
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#11524

Subgroups

	subgroup	G/H
center	C₄×C₅	#410
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×M₁₆	C₂³×C₅
socle	C₂×C₅	#1464
prime radical	C₂×C₅×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₄²×C₅ ◁ C₄×C₅×M₁₆ ◁ G

288 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 98 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)(6,7,10,16,11,15,26,32,12,17,25,33,9,14,23,31)(8,18,21,34,20,13,27,29,19,28,24,36,22,30,35,37);(6,8)(7,13)(9,22)(10,24)(11,20)(12,19)(14,18)(15,28)(16,29)(17,30)(21,25)(23,27)(26,35)(31,34)(32,36)(33,37);(7,14,17,15)(8,19)(10,25)(13,28,30,18)(16,32,33,31)(20,22)(23,26)(29,34,37,36)

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, 4512);