C₅×#12837 #11935557

order	1280
nilpotency class	4
isoclinism	isoclinic class of #2215
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#12837

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂²×C₅	C₂×#107
prime radical	C₂⁴×C₅	Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂³×C₄×C₅ ◁ C₂³×C₅×C₈ ◁ G

394 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 166 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) = 41
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 41 points:

(1,2,3,4,5)(6,7)(8,9)(10,11,14,19,16,20,15,30)(12,21,24,36,26,37,25,40)(13,27,28,34,29,41,22,18)(17,31,32,39,33,38,35,23);(10,12,22,38,15,25,29,39,16,26,28,31,14,24,13,23)(11,17,27,36,19,32,34,37,20,33,41,40,30,35,18,21);(7,9)

Conjugacy classes & character table

380 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5825);