C₅×#10045 #11932765

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10038
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#10045

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂³×C₅	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

444 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 53
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 3 generators on 53 points:

(1,2,3,4,5)(6,7)(8,14)(9,17)(10,13)(11,16)(12,19)(15,21)(18,20)(22,23,28,33)(24,34,37,45)(25,30,41,48)(26,32,27,31)(29,38,44,36)(35,50,51,46)(39,53,52,43)(40,49,42,47);(11,20)(14,21)(22,24,28,37)(23,29,33,44)(25,39,41,52)(26,38,27,36)(30,46,48,50)(31,45,32,34)(35,42,51,40)(43,49,53,47);(6,8)(7,11)(9,18)(10,15)(12,21)(13,20)(14,17)(16,19)(22,25,28,41)(23,30,33,48)(24,35,37,51)(26,40,27,42)(29,43,44,53)(31,47,32,49)(34,46,45,50)(36,39,38,52)

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, 3033);