C₅²×#455 #13103192

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×#455

Subgroups

	subgroup	G/H
center, socle	C₂×C₅²	#120
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₅²
prime radical	C₅²×H(5,𝔽₂)	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅² ◁ C₂×C₄×C₅² ◁ G

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

normal subgroups only

This group has 608 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)(11,12,15,18)(13,19,21,26)(14,22,24,17)(16,23,25,20);(6,7,8,9,10)(11,13)(12,16)(14,23)(15,21)(17,26)(18,25)(19,22)(20,24);(14,24)(17,22)(19,26)(20,23)

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, 5544);