C₅×#12924 #11935644

order	1280
nilpotency class	4
isoclinism	isoclinic class of #2190
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#12924

Subgroups

	subgroup	G/H
center	C₄×C₅	#447
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂×C₅	#1852
prime radical	C₂⁴×C₅	Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂²×C₅×M₁₆ ◁ G

386 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 146 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)(6,7,10,15,12,16,11,26)(8,17,20,34,19,33,27,31)(9,21,24,32,23,36,25,14)(13,22,29,35,28,37,30,18);(6,8,10,20,12,19,11,27)(7,13,15,29,16,28,26,30)(9,22,24,35,23,37,25,18)(14,31,21,17,32,34,36,33);(9,23)(14,32)(17,33)(18,35)(21,36)(22,37)(24,25)(31,34)

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, 5912);