C₅×#11458 #11934178

order	1280
nilpotency class	4
isoclinism	isoclinic class of #2121
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅×C₈ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#11458

Subgroups

	subgroup	G/H
center	C₄×C₅	C₂×#99
commutator	C₂³	C₂²×C₅×C₈
Frattini, cosocle	C₂×M₁₆	C₂³×C₅
socle	C₂×C₅	C₂×#324
prime radical	C₂⁴×C₅	C₂×C₈
LCS	G ▷ C₂³ ▷ 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)(6,7,9,14,10,15,20,28,11,13,19,25,21,26,32,35)(8,12,17,23,18,24,31,34,16,22,29,27,30,33,37,36);(6,8)(7,12,13,22)(9,17)(10,18)(11,16)(14,27,25,23)(15,24,26,33)(19,29)(20,31)(21,30)(28,36,35,34)(32,37);(8,16)(12,22)(17,29)(18,30)(23,27)(24,33)(31,37)(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, 4446);