C₅×#14711 #11937431

order	1280
nilpotency class	2
isoclinism	isoclinic class of #2846
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₅ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

direct product

C₅×#14711

Subgroups

	subgroup	G/H
center, socle	C₂⁴×C₅	C₂⁴
commutator	C₂³	C₂³×C₄×C₅
Frattini, cosocle	C₂⁴	C₂⁴×C₅
prime radical	C₂⁵×C₅	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅ ◁ G

1770 conjugacy classes of subgroups; only the 542 normal subgroups are shown.

normal subgroups only

This group has 1770 conjugacy classes of subgroups — too many to draw. Only its 542 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 — 4 generators on 29 points:

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

Conjugacy classes & character table

440 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 7699);