C₅×#24210 #11946930

order	1280
nilpotency class	3
isoclinism	isoclinic class of #23918
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁶

Aliases

	subgroup	G/H
center, socle	C₂³×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂²×C₄²×C₅ ◁ G

1566 conjugacy classes of subgroups.

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 4 generators on 21 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 17198);