C₅×#18731 #11941451

order	1280
nilpotency class	3
isoclinism	isoclinic class of #18712
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#18731

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅
prime radical	C₂×C₅×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

884 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 884 conjugacy classes of subgroups — too many to draw. Only its 230 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) = 320

A minimal faithful permutation representation — 4 generators on 37 points:

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

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 11719);