C₅×#12031 #11934751

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12028
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₅×#12031

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#106
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₄×C₈	C₂³×C₅
prime radical	C₂³×C₅	#106
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

200 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 53
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 3 generators on 53 points:

(1,2,3,4,5)(6,7,9,12,10,13,16,19)(8,11,15,18,14,17,20,21)(22,23)(24,33,36,48)(25,30)(26,39)(27,42)(28,32)(29,35,44,51)(31,40)(34,47,37,49)(38,46)(41,53)(43,50,45,52);(6,8)(7,11)(9,15)(10,14)(12,18)(13,17)(16,20)(19,21)(22,24,26,35,27,34,40,50,28,36,41,51,25,37,38,52)(23,29,31,33,30,45,39,49,32,44,46,48,42,43,53,47);(8,14)(11,17)(15,20)(18,21)(22,25,28,27)(23,30,32,42)(24,34,36,37)(26,38,41,40)(29,43,44,45)(31,39,46,53)(33,47,48,49)(35,50,51,52)

Conjugacy classes & character table

230 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5019);