C₅²×#456 #13103193

order	1600
nilpotency class	3
isoclinism	isoclinic class of #453
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅² · Schur multiplier (H₂) = C₂³×C₅

Aliases

direct product

C₅²×#456

Subgroups

	subgroup	G/H
center, socle	C₂×C₅²	#120
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₅²
prime radical	C₅²×2₋¹⁺⁴	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅² ◁ C₂×C₄×C₅² ◁ G

568 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 216 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)(11,12,16,14)(13,19,22,26)(15,23,25,18)(17,24,20,21);(6,7,8,9,10)(11,13)(12,17)(14,20)(15,24)(16,22)(18,26)(19,23)(21,25);(11,14,16,12)(13,20,22,17)(15,18,25,23)(19,24,26,21)

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, 5545);