C₅²×#510 #13103247

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×#510

Subgroups

	subgroup	G/H
center, socle	C₂×C₅²	C₂×D₈
commutator, Frattini, cosocle	C₈	C₂³×C₅²
prime radical	C₅²×#135	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅² ◁ C₂×C₄×C₅² ◁ C₂×C₅²×C₈ ◁ G

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

normal subgroups only

This group has 328 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) = 42
transitive degree	μ_t(G) = 800

A minimal faithful permutation representation — 3 generators on 42 points:

(1,2,3,4,5)(11,12)(13,21)(14,25)(15,30)(16,18)(17,22)(19,28)(20,27)(23,41)(24,38)(26,36)(29,37)(31,35)(32,40)(33,42)(34,39);(6,7,8,9,10)(11,13,16,24)(12,17,18,32)(14,26,29,42)(15,31,27,23)(19,35,34,41)(20,36,30,33)(21,37,38,25)(22,39,40,28);(12,18)(17,32)(19,34)(20,30)(21,38)(25,37)(33,36)(35,41)

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, 5599);