C₅×#490 #65653

order	320
nilpotency class	3
isoclinism	isoclinic class of #468
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₄

Aliases

	subgroup	G/H
center, socle	C₂²×C₅	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₅
prime radical	C₅×#127	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ G

110 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 3 generators on 21 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 991);