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C₂×C₅³ #6960

order	250
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) order 1488000
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₅³

Aliases

C₂×C₅³

Subgroups

	subgroup	G/H
center	G	C₁
commutator, Frattini	C₁	G
LCS	G ▷ C₁
UCS	C₁ ◁ G

128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 17 points:

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

Conjugacy classes & character table

250 conjugacy classes — view the full character table.

Relationships

center of	C₅³×D₄, C₅³×Q₈, C₅³×Dic₃, …
derived subgroup of	#11758752, #11758753, #11758754, …
Fitting subgroup of	C₅²×Dic₅, C₅×#1039, #92553, …
socle of	C₅²×Dic₅, C₅×#1039, #92553, …
maximal subgroup of	C₅²×Dic₅, C₅×#1039, #92553, …
maximal normal subgroup of	C₅²×Dic₅, C₅×#1039, #92553, …
maximal quotient of	C₄×C₅³, C₂²×C₅³, C₅³×D₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(250, 15);