C₅×#8734 #11931454

order	1280
nilpotency class	3
isoclinism	isoclinic class of #1828
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅×C₈ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#8734

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#381
commutator	C₂³	C₂²×C₅×C₈
Frattini, cosocle	C₂³×C₄	C₂³×C₅
prime radical	C₂⁵×C₅	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₄×C₅ ◁ G

622 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 622 conjugacy classes of subgroups — too many to draw. Only its 166 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) = 160

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

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

Conjugacy classes & character table

290 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 1722);