C₅×#7256 #11929976

order	1280
nilpotency class	4
isoclinism	isoclinic class of #1339
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#7256

Subgroups

	subgroup	G/H
center	C₄×C₅	#327
commutator	C₂×C₈	C₄²×C₅
Frattini, cosocle	C₂²×M₁₆	C₂²×C₅
socle	C₂×C₅	#1294
prime radical	C₂⁴×C₅	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂²×C₅×M₁₆ ◁ G

318 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 98 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)(6,7,10,15,12,16,11,27)(8,17,19,32,21,34,20,35)(9,22,24,33,25,37,26,14)(13,28,29,36,31,23,30,18);(6,8)(7,13)(9,23,25,28)(10,19)(11,20)(12,21)(14,32,33,35)(15,29)(16,31)(17,22,34,37)(18,26,36,24)(27,30)

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, 244);