C₂×C₅×AGL(1,ℤ/8) #66237

order	320
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

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

298 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 162 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 4 generators on 15 points:

(1,2)(3,4,5,6,7);(8,9)(10,12)(11,13)(14,15);(9,12)(10,15)(13,14);(1,2)(9,13)(12,14)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1575);