C₂×C₅×Q₈² #10619262

order	640
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₅ · Schur multiplier (H₂) = C₂⁸

Aliases

C₂×C₅×Q₈²

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₅
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ G

1112 conjugacy classes of subgroups.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 5 generators on 23 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 21344);