C₂³×C₅×Q₁₆ #10619361

order	640
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₅ · Schur multiplier (H₂) = C₂⁹

Aliases

C₂³×C₅×Q₁₆

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₅	D₄
commutator, Frattini, cosocle	C₄	C₂⁵×C₅
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅ ◁ C₂³×C₄×C₅ ◁ G

1464 conjugacy classes of subgroups.

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 5 generators on 27 points:

(1,2)(23,24,25,26,27);(3,4);(5,6);(7,8,9,10,11,12,13,14)(15,22,21,20,19,18,17,16);(7,15,11,19)(8,16,12,20)(9,17,13,21)(10,18,14,22)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 21443);