C₅×#12418 #11935138

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12416
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#12418

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#459
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₅
prime radical	C₅×#2206	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×C₈² ◁ G

256 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 82 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)(7,10)(8,12)(11,13)(14,15)(16,19)(17,20)(18,26)(21,25)(22,24)(23,27)(28,29);(6,7,8,10,9,11,12,13)(15,19)(20,27)(21,28)(22,26);(15,20)(17,25)(18,24)(19,27)(21,22)(23,29)(26,28)

Conjugacy classes & character table

215 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5406);