C₅×#10586 #11933306

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10078
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#10586

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ G

350 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 114 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)(6,7,10,12)(8,13,9,11)(14,15,19,23)(16,20,26,29)(17,21,18,22)(24,27,25,28)(30,31)(32,39)(33,37)(34,36)(35,40)(38,42)(41,45)(43,44);(6,8,10,9)(7,11,12,13)(14,16)(15,20)(17,25)(18,24)(19,26)(21,28)(22,27)(23,29)(30,32,33,40)(31,35,37,39)(34,43,42,41)(36,45,38,44);(6,9,10,8)(7,11,12,13)(15,21)(16,24)(17,18)(22,23)(25,26)(27,28)(31,36)(34,42)(35,44)(37,38)(39,45)(41,43)

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, 3574);