C₅×#24176 #11946896

order	1280
nilpotency class	3
isoclinism	isoclinic class of #23906
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

1238 conjugacy classes of subgroups.

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 160

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 17164);