C₅×#12298 #11935018

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12298
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

	subgroup	G/H
center, socle	C₂×C₅	#1613
commutator, Frattini, cosocle	C₂×M₁₆	C₂³×C₅
prime radical	C₅×#3192	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×M₁₆ ◁ G

376 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 70 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 — 3 generators on 21 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5286);