C₅×#9028 #11931748

order	1280
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₅×#9028

Subgroups

	subgroup	G/H
center, socle	C₂³×C₅	#120
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
prime radical	C₂⁵×C₅	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂³×C₄×C₅ ◁ G

696 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 33
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 33 points:

(1,2,3,4,5)(7,9)(10,11,15,16,14,12,13,17)(18,19)(20,23)(21,28,29,24)(22,25)(26,30,32,33)(27,31);(6,7,8,9)(10,12,14,11)(13,17,15,16)(18,20,22,27)(19,23,25,31)(21,26,29,32)(24,30,28,33);(10,13,14,15)(11,16,12,17)(18,21,22,29)(19,24,25,28)(20,26,27,32)(23,30,31,33)

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, 2016);