C₅×#7391 #11930111

order	1280
nilpotency class	5
isoclinism	isoclinic class of #7389
automorphism	?
homology	Abelianization (H₁) = C₂×C₅×C₈ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#7391

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1333
commutator	C₁₆	C₂×C₅×C₈
Frattini, cosocle	#346	C₂²×C₅
prime radical	C₂²×C₅	#361
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₄×C₅×C₈ ◁ C₅×#346 ◁ G

116 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 116 conjugacy classes of subgroups — too many to draw. Only its 50 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 — 2 generators on 69 points:

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

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 379);