C₅×#12088 #11934808

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12087
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#12088

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₅	#442
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₄×C₈	C₂³×C₅
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#322 ◁ G

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

normal subgroups only

This group has 198 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) = 1280

A minimal faithful permutation representation — 3 generators on 45 points:

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

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5076);