C₅×#8507 #11931227

order	1280
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#8507

Subgroups

	subgroup	G/H
center, socle	C₂³×C₅	#120
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂²×C₅×D₄	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ G

736 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 158 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) = 33
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 3 generators on 33 points:

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

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, 1495);