C₅×#8463 #11931183

order	1280
nilpotency class	3
isoclinism	isoclinic class of #8453
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#8463

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#379
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
prime radical	C₅×#3493	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₄×C₅ ◁ G

718 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 718 conjugacy classes of subgroups — too many to draw. Only its 142 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)(8,14,16,11)(9,12)(10,13)(15,20,21,18)(17,19)(22,23,27,32,28,33,26,31)(24,34,38,29,37,44,36,43)(25,30,39,45,40,46,41,47)(35,49,50,53,51,48,52,42);(6,8)(7,11)(9,15)(10,16)(12,18)(13,14)(17,21)(19,20)(22,24)(23,29)(25,35)(26,38)(27,36)(28,37)(30,42)(31,44)(32,34)(33,43)(39,52)(40,51)(41,50)(45,48)(46,53)(47,49);(6,9,10,17)(7,12,13,19)(8,15,16,21)(11,18,14,20)(22,25)(23,30)(24,35)(26,41)(27,39)(28,40)(29,42)(31,47)(32,45)(33,46)(34,48)(36,52)(37,51)(38,50)(43,53)(44,49)

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 1451);