C₂²×5₋¹⁺² #92550

order	500
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₅)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅² · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂²×5₋¹⁺²
unambiguous semidirect product	(C₂²×C₂₅)⋊C₅

	subgroup	G/H
center, socle	C₂²×C₅	C₅²
commutator, Frattini, cosocle	C₅	C₂²×C₅²
prime radical	C₂²×C₅²	C₅
LCS	G ▷ C₅ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ G

50 conjugacy classes of subgroups.

minimal degree	μ(G) = 29
transitive degree	μ_t(G) = 100

A minimal faithful permutation representation — 2 generators on 29 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(500, 36);