C₅×#11456 #11934176

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×#11456

Subgroups

	subgroup	G/H
center	C₄×C₅	C₂×#99
commutator	C₂×C₄	C₂²×C₅×C₈
Frattini, cosocle	C₂×M₁₆	C₂³×C₅
socle	C₂×C₅	C₂×#324
prime radical	C₂⁴×C₅	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂²×C₅×M₁₆ ◁ G

364 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 138 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)(6,7,10,15,11,14,25,30,12,16,26,32,9,22,23,31)(8,17,20,34,21,28,35,37,18,33,24,36,19,13,27,29);(6,8)(7,13,16,28)(9,19)(10,24)(11,21)(12,18)(14,17,22,33)(15,29,32,37)(20,26)(23,35)(25,27)(30,34,31,36);(8,18)(13,28)(17,33)(19,21)(20,24)(27,35)(29,37)(34,36)

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, 4444);