C₅²×#509 #13103246

order	1600
nilpotency class	4
isoclinism	isoclinic class of #509
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅² · Schur multiplier (H₂) = C₂²×C₅

Aliases

direct product

C₅²×#509

Subgroups

	subgroup	G/H
center, socle	C₂×C₅²	C₂×D₈
commutator, Frattini, cosocle	C₈	C₂³×C₅²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅² ◁ C₂×C₄×C₅² ◁ C₂×C₅²×C₈ ◁ G

360 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 184 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)(13,16)(14,23)(15,17)(18,20)(19,21)(22,24)(25,26);(6,7,8,9,10)(12,16)(13,20)(14,21)(17,24)(18,19)(22,26)(23,25);(12,17)(16,24)(18,25)(19,23)

Conjugacy classes & character table

400 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₅⋊D₅×#509, #509×#1043, #72253107, …
maximal subgroup of	C₅⋊D₅×#509, C₅²×AGL(1,ℤ/16), #72253107, …
maximal normal subgroup of	C₅⋊D₅×#509, C₅²×AGL(1,ℤ/16), #72253107, …
maximal quotient of	C₅²×#2177, C₂×C₅²×#509

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1600, 5598);