C₅×#9985 #11932705

order	1280
nilpotency class	3
isoclinism	isoclinic class of #9980
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#9985

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#392
commutator, Frattini, cosocle	C₂³×C₄	C₂³×C₅
prime radical	C₂⁴×C₅	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₄×C₅ ◁ G

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

normal subgroups only

This group has 502 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) = 320

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

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

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 2973);