order	250
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₅)
automorphism	Aut(G) = AGL(2,𝔽₅) (order 12000) · Inn(G) = C₅² · Out(G) = GL(2,𝔽₅)
homology	Abelianization (H₁) = C₂×C₅² · Schur multiplier (H₂) = C₅²

Aliases

direct product	C₂×H(3,𝔽₅)
unambiguous semidirect product	(C₂×C₅²)⋊C₅

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	C₅²
commutator, Frattini, cosocle	C₅	C₂×C₅²
LCS	G ▷ C₅ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ G

30 conjugacy classes of subgroups.

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 50

A minimal faithful permutation representation — 2 generators on 27 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(250, 10);