C₅×#13487 #11936207

order	1280
nilpotency class	4
isoclinism	isoclinic class of #13398
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#13487

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#481
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂²×C₅×D₈	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#97 ◁ G

338 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 106 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)(6,7,8,9)(10,11,14,17,16,19,13,15)(12,20,21,25,23,24,22,18);(11,17)(12,18)(13,14)(15,19)(20,22)(21,24)(23,25);(7,9)(10,12,13,21,16,23,14,22)(11,18,15,20,19,25,17,24)

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 6475);