C₅×#12027 #11934747

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12026
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#12027

Subgroups

	subgroup	G/H
center	C₄×C₅	#427
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₄×C₈	C₂³×C₅
socle	C₂×C₅	#1568
prime radical	C₂³×C₅	#108
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₅×#97 ◁ C₅×#405 ◁ G

180 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 69 points:

(1,2,3,4,5)(6,7,11,18,13,19,37,50,14,20,36,17,12,40,10,35)(8,21,24,42,26,52,23,43,27,48,38,66,25,60,58,15)(9,28,31,62,33,47,30,49,34,63,39,67,32,16,44,46)(22,53,55,41,57,68,65,69,29,64,59,45,56,51,54,61);(6,8,10,23,12,25,36,24,14,27,37,58,13,26,11,38)(7,15,17,21,19,42,35,52,20,43,18,48,40,66,50,60)(9,29,30,65,32,57,31,55,34,22,44,54,33,56,39,59)(16,45,46,68,28,61,62,64,47,41,49,51,63,69,67,53);(6,9,12,32,14,34,13,33)(7,16,19,28,20,47,40,63)(8,22,25,56,27,29,26,57)(10,30,36,31,37,44,11,39)(15,41,42,69,43,45,66,61)(17,46,35,62,18,49,50,67)(21,51,52,53,48,68,60,64)(23,54,24,59,58,65,38,55)

Conjugacy classes & character table

260 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5015);