order	500
isoclinism	isoclinic class of C₅⋊(C₅⋊D₅)
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂×C₅³

Aliases

direct product	C₂×C₅⋊(C₅⋊D₅)
unambiguous semidirect product	(C₂×C₅)⋊(C₅⋊D₅)

Splittings

Fitting, socle	C₂×C₅³ ⋊ C₂
derived, Sylow	C₅³ ⋊ C₂²

	subgroup	G/H
center, hypercenter	C₂	C₅⋊(C₅⋊D₅)
commutator, cosocle, hypocenter	C₅³	C₂²
Frattini	C₁	G
Fitting, socle	C₂×C₅³	C₂
LCS	G ▷ C₅³
UCS	C₁ ◁ C₂

320 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 250

A minimal faithful permutation representation — 4 generators on 17 points:

(1,2)(4,5)(6,7)(9,10)(11,12)(14,15)(16,17);(1,2)(3,4,6,7,5);(8,9,11,12,10);(13,14,16,17,15)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(500, 55);