C₅×#12080 #11934800

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12077
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#12080

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#442
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₄×C₈	C₂³×C₅
prime radical	C₂×C₅×D₈	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#322 ◁ G

218 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 29
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 3 generators on 29 points:

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

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, 5068);