C₅×#7851 #11930571

order	1280
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₅ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₅×#7851

Subgroups

	subgroup	G/H
center	C₂×C₄²×C₅	C₂³
commutator	C₂³	C₂×C₄²×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂³×C₅	C₂×C₄²
prime radical	C₂⁴×C₅	C₄²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₅ ◁ G

610 conjugacy classes of subgroups; only the 286 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 286 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) = 41
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 3 generators on 41 points:

(1,2,3,4,5)(6,7)(8,9)(10,11,13,17,14,16,20,23)(12,15,19,21,18,22,24,25)(26,27)(28,31)(29,33)(30,32)(34,38)(35,37)(36,39)(40,41);(7,9)(10,12)(11,15)(13,19)(14,18)(16,22)(17,21)(20,24)(23,25)(26,28,29,34,30,35,36,40)(27,31,33,38,32,37,39,41);(11,16)(12,18)(17,23)(19,24)(27,32)(31,37)(33,39)(38,41)

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, 839);