C₅×#11863 #11934583

order	1280
nilpotency class	4
isoclinism	isoclinic class of #11855
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#11863

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

304 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 118 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)(22,23,27,33,29,35,28,34)(24,36,39,46,41,51,40,50)(25,42,26,45,44,32,43,31)(30,37,47,38,49,53,48,52);(6,8)(7,11)(9,17)(10,16)(12,20)(13,14)(15,18)(19,21)(22,24,29,41)(23,30,35,49)(25,38,44,52)(26,37,43,53)(27,40,28,39)(31,36,45,51)(32,46,42,50)(33,48,34,47);(6,9,10,18)(7,12,13,21)(8,15,16,17)(11,19,14,20)(22,25,28,43,29,44,27,26)(23,31,33,42,35,45,34,32)(24,37,40,52,41,53,39,38)(30,46,47,51,49,50,48,36)

Conjugacy classes & character table

260 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 4851);