C₅²×#489 #13103226

order	1600
nilpotency class	3
isoclinism	isoclinic class of #468
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅² · Schur multiplier (H₂) = C₄×C₅

Aliases

direct product

C₅²×#489

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅²	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₅²
prime radical	C₂×C₅²×D₄	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅² ◁ C₄²×C₅² ◁ G

400 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 200 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) = 26
transitive degree	μ_t(G) = 800

A minimal faithful permutation representation — 3 generators on 26 points:

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

Conjugacy classes & character table

400 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1600, 5578);