C₅×#7303 #11930023

order	1280
nilpotency class	4
isoclinism	isoclinic class of #7262
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#7303

Subgroups

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

204 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 90 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)(14,15,18,24,21,26,20,36)(16,27,29,42,31,43,34,38)(17,25,19,35,33,41,32,23)(22,30,37,28,40,44,39,45);(6,8,11,7)(9,12,10,13)(14,16,19,30,18,34,33,45,21,31,32,44,20,29,17,28)(15,22,25,38,36,37,23,27,26,40,41,42,24,39,35,43)

Conjugacy classes & character table

230 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 291);