C₅×#7360 #11930080

order	1280
nilpotency class	5
isoclinism	isoclinic class of #1412
automorphism	?
homology	Abelianization (H₁) = C₂×C₅×C₈ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#7360

Subgroups

	subgroup	G/H
center	C₄×C₅	#354
commutator	C₄²	C₂×C₅×C₈
Frattini, cosocle	#423	C₂²×C₅
socle	C₂×C₅	#1319
prime radical	C₂³×C₅	#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#423 ◁ G

164 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 46 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)(6,7,9,14,10,15,24,33,11,16,25,34,26,13,23,31)(8,17,19,35,21,12,27,28,18,29,36,32,20,30,22,37);(6,8,11,18)(7,12)(9,22,25,27)(10,21,26,20)(13,17)(14,32)(15,29)(16,30)(19,23,36,24)(28,31)(33,37)(34,35)

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, 348);