C₅×#29658 #11952378

order	1280
nilpotency class	3
isoclinism	isoclinic class of #29634
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₅×#29658

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₅
prime radical	C₂⁵×C₅	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₄×C₅ ◁ G

1086 conjugacy classes of subgroups; only the 238 normal subgroups are shown.

normal subgroups only

This group has 1086 conjugacy classes of subgroups — too many to draw. Only its 238 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 — 4 generators on 37 points:

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

Conjugacy classes & character table

185 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 22646);