C₅×#25176 #11947896

order	1280
nilpotency class	3
isoclinism	isoclinic class of #23860
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

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

1262 conjugacy classes of subgroups.

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 320

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 18164);