C₅×#11486 #11934206

order	1280
nilpotency class	4
isoclinism	isoclinic class of #2127
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅×C₈ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#11486

Subgroups

	subgroup	G/H
center	C₄×C₅	C₂×#99
commutator	C₂×C₄	C₂²×C₅×C₈
Frattini, cosocle	C₂×M₁₆	C₂³×C₅
socle	C₂×C₅	#1459
prime radical	C₅×#532	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₅×C₈ ◁ C₂×C₅×#131 ◁ G

364 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 126 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)(6,7,11,17,13,18,25,37,12,19,29,30,10,16,21,36)(8,20,23,33,22,32,9,15,24,34,27,35,28,14,26,31);(6,8)(7,14)(9,25)(10,28)(11,23)(12,24)(13,22)(15,17)(16,34)(18,20)(19,32)(21,26)(27,29)(30,31)(33,36)(35,37);(6,9,13,27,12,26,10,23)(7,15,18,35,19,31,16,33)(8,21,22,11,24,25,28,29)(14,30,20,36,32,17,34,37)

Conjugacy classes & character table

230 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 4474);