C₅×#9018 #11931738

order	1280
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₅×#9018

Subgroups

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

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

normal subgroups only

This group has 928 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) = 25
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 25 points:

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

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, 2006);