C₅×#8540 #11931260

order	1280
nilpotency class	3
isoclinism	isoclinic class of #8534
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#8540

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#379
commutator	C₂⁴	C₂²×C₄×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
prime radical	C₅×#534	C₄
LCS	G ▷ C₂⁴ ▷ 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)(22,23,26,30,27,31,37,43)(24,28,34,40,33,39,47,51)(25,29,36,42,35,41,48,52)(32,44,46,53,45,38,49,50);(6,8)(7,11)(9,15)(10,16)(12,18)(13,14)(17,21)(19,20)(22,24)(23,28)(25,32)(26,34)(27,33)(29,38)(30,40)(31,39)(35,45)(36,46)(37,47)(41,44)(42,50)(43,51)(48,49)(52,53);(6,9,10,17)(7,12,13,19)(8,15,16,21)(11,18,14,20)(22,25)(23,29)(24,32)(26,36)(27,35)(28,38)(30,42)(31,41)(33,45)(34,46)(37,48)(39,44)(40,50)(43,52)(47,49)(51,53)

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, 1528);