C₅×#29657 #11952377

order	1280
nilpotency class	3
isoclinism	isoclinic class of #29634
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#29657

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 ◁ C₂⁵×C₅ ◁ G

1148 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 1148 conjugacy classes of subgroups — too many to draw. Only its 242 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)(9,13,20,26)(11,15,22,28)(17,31,32,25)(19,30,34,24)(21,27,35,36)(23,33,29,37);(6,8,10,18)(7,12,14,16)(9,17,20,32)(11,19,22,34)(13,25,26,31)(15,24,28,30)(21,33,35,37)(23,36,29,27);(6,9,11,21)(7,13,15,27)(8,17,19,33)(10,20,22,35)(12,23,24,25)(14,26,28,36)(16,29,30,31)(18,32,34,37);(9,20)(13,26)(17,32)(21,35)(23,29)(25,31)(27,36)(33,37)

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, 22645);