C₅×#11496 #11934216

order	1280
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#11496

Subgroups

	subgroup	G/H
center	C₂×C₅×C₈	C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₅×C₈
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂²×C₅	C₂×C₂²⋊C₈
prime radical	C₂³×C₅	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₅×C₈ ◁ C₂×C₄×C₅×C₈ ◁ G

296 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 138 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)(15,20,21,18)(17,19)(22,23,27,33,29,32,41,49,28,34,40,50,26,42,25,31)(24,35,38,45,37,30,44,46,39,48,52,53,43,47,36,51);(6,8,10,16)(7,11,13,14)(9,15,17,21)(12,18,19,20)(22,24,28,39)(23,30,34,47)(25,36,41,44)(26,43,29,37)(27,38,40,52)(31,45,49,53)(32,48,42,35)(33,46,50,51);(6,9,10,17)(7,12,13,19)(8,15,16,21)(11,18,14,20)(22,25,26,40,28,41,29,27)(23,31,42,50,34,49,32,33)(24,36,43,52,39,44,37,38)(30,45,35,51,47,53,48,46)

Conjugacy classes & character table

380 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 4484);