C₅²×#472 #13103209

order	1600
nilpotency class	3
isoclinism	isoclinic class of #471
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅² · Schur multiplier (H₂) = C₂²×C₅

Aliases

direct product

C₅²×#472

Subgroups

	subgroup	G/H
center, socle	C₂×C₅²	#121
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₅²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅² ◁ C₂×C₄×C₅² ◁ G

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

normal subgroups only

This group has 432 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) = 400

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

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

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, 5561);