C₅×#8791 #11931511

order	1280
nilpotency class	3
isoclinism	isoclinic class of #8790
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#8791

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#382
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂³×C₅	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

262 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 320

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

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

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, 1779);