C₅×#12059 #11934779

order	1280
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#12059

Subgroups

	subgroup	G/H
center	C₂×C₅×C₈	C₂×D₄
commutator	C₄	C₂²×C₅×C₁₆
Frattini, cosocle	C₄×C₈	C₂³×C₅
socle, prime radical	C₂²×C₅	C₈×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅×C₈ ◁ C₄×C₅×C₁₆ ◁ G

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

normal subgroups only

This group has 212 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) = 37
transitive degree	μ_t(G) = 1280

A minimal faithful permutation representation — 3 generators on 37 points:

(1,2,3,4,5)(6,7)(8,14)(9,17)(10,12)(11,15)(13,19)(16,21)(18,20)(22,23,25,29,26,30,33,36,27,24,28,31,34,32,35,37);(6,8,10,16)(7,11,12,20)(9,18,19,15)(13,21,17,14);(7,12)(11,20)(13,17)(14,21)(22,24,25,31,26,32,33,37,27,23,28,29,34,30,35,36)

Conjugacy classes & character table

440 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5047);