C₅×#8545 #11931265

order	1280
nilpotency class	3
isoclinism	isoclinic class of #8541
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₅×#8545

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#379
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

910 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 910 conjugacy classes of subgroups — too many to draw. Only its 150 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)(8,16,19,12)(9,21,22,13)(10,14)(11,15)(17,24)(18,32,33,25)(20,29,30,26)(23,37,31,27)(28,34,36,35);(6,8,17,30)(7,12,24,29)(9,18,31,33)(10,19,11,20)(13,25,37,32)(14,16,15,26)(21,36,27,28)(22,35,23,34);(6,9,10,23)(7,13,14,27)(8,18,19,34)(11,22,17,31)(12,25,16,28)(15,21,24,37)(20,35,30,33)(26,36,29,32)

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, 1533);