C₅×#12321 #11935041

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12316
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#12321

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#447
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₅
prime radical	C₅×#3287	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×C₈×D₄ ◁ G

384 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 86 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)(6,7)(8,11)(9,12)(10,13)(14,15,20,24)(16,25,28,43)(17,22,19,34)(18,30,33,44)(21,36,39,27)(23,29,41,32)(26,42,35,40)(31,37,45,38);(7,11)(8,9)(12,13)(14,16)(15,21)(17,26)(18,31)(19,35)(20,28)(22,37)(23,40)(24,39)(25,30)(27,32)(29,36)(33,45)(34,38)(41,42)(43,44);(7,12)(8,9)(11,13)(14,17,20,19)(15,22,24,34)(16,26,28,35)(18,29,33,32)(21,37,39,38)(23,30,41,44)(25,42,43,40)(27,31,36,45)

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5309);