C₂₅ #75

order	25
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₄×C₅ (order 20) · Inn(G) = C₁ · Out(G) = C₄×C₅
holomorph	G ⋊ Aut(G) = AGL(1,ℤ/25)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

C₂₅

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

3 conjugacy classes of subgroups.

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 25

A minimal faithful permutation representation — 1 generator on 25 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)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(25, 1);