C₅×#7477 #11930197

order	1280
nilpotency class	5
isoclinism	isoclinic class of #1418
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#7477

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#357
commutator	C₁₆	C₄²×C₅
Frattini, cosocle	C₂²×C₁₆	C₂²×C₅
prime radical	C₂³×C₅	D₁₆
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂²×C₅×C₈ ◁ C₂²×C₅×C₁₆ ◁ G

178 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 102 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 — 2 generators on 41 points:

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

Conjugacy classes & character table

260 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 465);