C₅×#7478 #11930198

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

Aliases

direct product

C₅×#7478

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#357
commutator	C₁₆	C₄²×C₅
Frattini, cosocle	C₂²×C₁₆	C₂²×C₅
prime radical	C₂³×C₅	SD₃₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂²×C₅×C₈ ◁ C₂²×C₅×C₁₆ ◁ G

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

normal subgroups only

This group has 174 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) = 640

A minimal faithful permutation representation — 2 generators on 41 points:

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

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, 466);