C₅×#8750 #11931470

order	1280
nilpotency class	3
isoclinism	isoclinic class of #8746
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₅×#8750

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#381
commutator	C₂⁴	C₂²×C₄×C₅
Frattini, cosocle	C₂⁵	C₂³×C₅
prime radical	C₂⁵×C₅	C₂³
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂⁵×C₅ ◁ G

538 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 110 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)(8,16,18,12)(9,20,22,26)(10,14)(11,15)(13,21,27,37)(17,31,33,29)(19,30,35,25)(23,28)(24,32,36,34);(6,8,11,19)(7,12,15,25)(9,17,22,34)(10,18,23,35)(13,24,27,31)(14,16,28,30)(20,36,26,29)(21,32,37,33);(6,9)(7,13,14,26)(8,17)(10,21)(11,22)(12,24,16,29)(15,27,28,20)(18,32)(19,34)(23,37)(25,31,30,36)(33,35)

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 1738);