C₅×#11865 #11934585

order	1280
nilpotency class	4
isoclinism	isoclinic class of #11858
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#11865

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂³×C₅	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

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

normal subgroups only

This group has 300 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) = 53
transitive degree	μ_t(G) = 640

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

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

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, 4853);