C₅×#12047 #11934767

order	1280
nilpotency class	4
isoclinism	isoclinic class of #509
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅×C₈ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#12047

Subgroups

	subgroup	G/H
center	C₅×C₈	C₂×D₈
commutator	C₈	C₂²×C₅×C₈
Frattini, cosocle	C₄×C₈	C₂³×C₅
socle	C₂×C₅	C₈×D₈
prime radical	C₅×#508	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₅×C₈ ◁ C₂×C₅×C₁₆ ◁ C₄×C₅×C₁₆ ◁ G

212 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 160

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

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

Conjugacy classes & character table

320 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5035);