C₅×#61321 #11984041

order	1280
nilpotency class	3
isoclinism	isoclinic class of D₄×AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₅ · Schur multiplier (H₂) = C₂⁸

Aliases

	subgroup	G/H
center, socle	C₂²×C₅	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄×C₅×Q₈ ◁ G

2126 conjugacy classes of subgroups.

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 5 generators on 21 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 54309);