C₅×#11497 #11934217

order	1280
nilpotency class	3
isoclinism	isoclinic class of #1510
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#11497

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	C₂×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×C₄×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

288 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 130 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)(22,23,27,33,29,32,42,31,28,34,41,50,26,43,25,40)(24,35,38,51,37,30,45,47,39,49,53,46,44,48,36,52);(6,8,10,16)(7,11,13,14)(9,15,17,21)(12,18,19,20)(22,24,28,39)(23,30,34,48)(25,36,42,45)(26,44,29,37)(27,38,41,53)(31,46,40,51)(32,49,43,35)(33,47,50,52);(6,9,10,17)(7,12,13,19)(8,15,16,21)(11,18,14,20)(22,25,26,41,28,42,29,27)(23,31,43,33,34,40,32,50)(24,36,44,53,39,45,37,38)(30,46,35,47,48,51,49,52)

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, 4485);