order	225
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×C₃×C₄×C₅ (order 120) · Inn(G) = C₁ · Out(G) = C₂×C₃×C₄×C₅
holomorph	G ⋊ Aut(G) = AGL(1,ℤ/9)×AGL(1,ℤ/25)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₂₂₅
abelian	C₉×C₂₅

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, socle, prime radical, cosocle	C₃×C₅	C₃×C₅
LCS	G ▷ C₁
UCS	C₁ ◁ G

9 conjugacy classes of subgroups.

minimal degree	μ(G) = 34
transitive degree	μ_t(G) = 225

A minimal faithful permutation representation — 1 generator on 34 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(225, 1);