C₅²×#494 #13103231

order	1600
nilpotency class	3
isoclinism	isoclinic class of #465
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅² · Schur multiplier (H₂) = C₂×C₄×C₅

Aliases

	subgroup	G/H
center, socle, prime radical	C₂²×C₅²	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₅²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅² ◁ C₄²×C₅² ◁ G

488 conjugacy classes of subgroups.

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 1600

A minimal faithful permutation representation — 3 generators on 30 points:

(1,2,3,4,5)(12,14)(15,16,19,22)(17,23,26,30)(18,27,24,21)(20,28,29,25);(6,7,8,9,10)(15,17,24,28,19,26,18,25)(16,20,27,30,22,29,21,23);(11,12,13,14)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1600, 5583);