C₅×#559 #66223

order	320
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

	subgroup	G/H
center, socle	C₂²×C₅	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₅
prime radical	C₅×#127	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ G

236 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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1561);