C₅×#12518 #11935238

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12503
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂×C₈

Aliases

direct product

C₅×#12518

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₅
prime radical	C₅×#2246	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×C₈² ◁ G

330 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 110 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 — 3 generators on 45 points:

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

Conjugacy classes & character table

350 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5506);