C₅×#18552 #11941272

order	1280
nilpotency class	3
isoclinism	isoclinic class of #18524
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#18552

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅
prime radical	C₂×C₅×#555	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

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

normal subgroups only

This group has 982 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)(6,7)(8,15)(9,13)(10,18)(11,14)(12,16)(17,20)(19,21)(22,23,27,30)(24,32,35,36)(25,31,26,29)(28,34,37,33);(6,8,11,17)(7,12,14,21)(9,16,10,19)(13,20,18,15)(22,24,27,35)(23,28,30,37)(25,34,26,33)(29,36,31,32);(6,9,11,10)(7,13,14,18)(8,16,17,19)(12,20,21,15)(23,29)(25,26)(28,36)(30,31)(32,37)(33,34);(7,14)(12,21)(13,18)(15,20)(23,30)(28,37)(29,31)(32,36)

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, 11540);