C₅×#12165 #11934885

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12108
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#12165

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂²×C₅×D₈	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#424 ◁ G

350 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 320

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

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

Conjugacy classes & character table

155 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5153);