C₅×#32875 #11955595

order	1280
nilpotency class	3
isoclinism	isoclinic class of #32875
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂³

Aliases

	subgroup	G/H
center, socle	C₂×C₅	#2842
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
prime radical	C₂³×C₅×D₄	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₄×C₅ ◁ G

712 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 186 normal subgroups are shown.

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

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 4 generators on 21 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 25863);