C₅×#8505 #11931225

order	1280
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#8505

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	#120
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂²×C₅	#379
prime radical	C₅×#3531	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ G

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

normal subgroups only

This group has 648 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)(7,11,13,16)(12,20,21,19)(22,23)(24,32,35,28)(25,36,37,47)(26,31)(27,30)(29,38,44,40)(33,49)(34,46,39,43)(41,45)(42,51)(48,52)(50,53);(6,7)(8,12)(9,16)(10,13)(11,18)(14,19)(15,21)(17,20)(22,24)(23,28)(25,33)(26,39)(27,35)(29,42)(30,32)(31,46)(34,41)(36,49)(37,50)(38,52)(40,51)(43,45)(44,48)(47,53);(6,8,10,15)(7,12,13,21)(9,17,18,14)(11,19,16,20)(22,25)(23,29)(24,33)(26,40)(27,37)(28,42)(30,44)(31,47)(32,48)(34,52)(35,50)(36,45)(38,41)(39,51)(43,49)(46,53)

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, 1493);