C₅×#3167 #10616762

order	640
nilpotency class	3
isoclinism	isoclinic class of #465
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

	subgroup	G/H
center	C₂×C₄×C₅	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₅
socle	C₂²×C₅	C₂²×D₄
prime radical	C₅×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ G

412 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)(6,7)(8,11)(9,10)(12,13)(15,18)(16,20)(19,21);(7,10)(11,13);(14,15,16,18,17,19,20,21);(6,8,9,12)(7,11,10,13)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 18844);