C₅×#13508 #11936228

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13331
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#13508

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#459
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂³×C₅	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

284 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 29
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 29 points:

(1,2,3,4,5)(6,7)(8,13)(9,11)(10,12)(14,15,17,21,19,22,27,18)(16,20,24,28,26,29,25,23);(6,8,10,9)(7,11,12,13)(14,16)(15,20)(17,24)(18,23)(19,26)(21,28)(22,29)(25,27);(6,9,10,8)(7,11,12,13)(15,18)(16,23)(17,27)(20,25)(21,22)(24,29)(26,28)

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, 6496);