C₅×#7297 #11930017

order	1280
nilpotency class	4
isoclinism	isoclinic class of #1347
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#7297

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	#102
commutator	C₂×C₈	C₄²×C₅
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₅
socle	C₂²×C₅	#340
prime radical	C₂³×C₅	#103
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

208 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 45
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 2 generators on 45 points:

(1,2,3,4,5)(6,7,10,12,11,8,9,13)(14,15,18,24,21,26,20,37)(16,27,29,39,32,41,31,45)(17,33,35,42,36,25,19,23)(22,30,38,34,40,44,43,28);(6,8,11,7)(9,12,10,13)(14,16,19,30,20,31,36,28,21,32,35,44,18,29,17,34)(15,22,25,39,24,38,23,41,26,40,33,45,37,43,42,27)

Conjugacy classes & character table

290 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 285);