C₅×#7484 #11930204

order	1280
nilpotency class	5
isoclinism	isoclinic class of #7481
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#7484

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1390
commutator	C₂×Q₈	C₄²×C₅
Frattini, cosocle	#532	C₂²×C₅
prime radical	C₅×#532	C₂²
LCS	G ▷ C₂×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#532 ◁ G

266 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 66 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 — 2 generators on 37 points:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 472);