C₂×C₅×#123 #66188

order	320
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

	subgroup	G/H
center, socle	C₂³×C₅	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁴×C₅
prime radical	C₂²×C₅×D₄	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ G

342 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 178 normal subgroups are shown.

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

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 4 generators on 19 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1526);