C₅×#7421 #11930141

order	1280
nilpotency class	5
isoclinism	isoclinic class of #7418
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#7421

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#357
commutator	C₂×C₁₆	C₂×C₄×C₅
Frattini, cosocle	C₂²×C₁₆	C₂²×C₅
prime radical	C₂³×C₅	SD₃₂
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂²×C₅×C₈ ◁ C₂²×C₅×C₁₆ ◁ G

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

normal subgroups only

This group has 140 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) = 77
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 2 generators on 77 points:

(1,2,3,4,5)(6,7)(8,12,13,10)(9,11)(14,15,18,24,21,28,44,27)(16,29,31,63,34,66,42,65)(17,35,38,72,41,58,77,73)(19,45,43,25,20,47,37,26)(22,49,50,30,54,67,53,69)(23,55,57,40,60,75,59,39)(32,70,46,64,33,71,48,56)(36,61,74,51,68,62,76,52);(6,8)(7,10)(9,13)(11,12)(14,16,21,34)(15,22,28,54)(17,36,41,68)(18,42,44,31)(19,46,20,48)(23,56,60,64)(24,53,27,50)(25,61,26,62)(29,35,66,58)(30,39,69,40)(32,37,33,43)(38,76,77,74)(45,51,47,52)(49,55,67,75)(57,71,59,70)(63,73,65,72)

Conjugacy classes & character table

245 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 409);