Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486312 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #62274 | C2 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #2929 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
690 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 690 conjugacy classes of subgroups — too many to draw. Only its 132 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 80 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 80 points:
(1,2,5,8)(3,6,10,13)(4,7,11,14)(9,15,16,12)(17,18)(19,33)(20,39)(21,29)(22,46)(23,31)(24,30)(25,32)(26,36)(27,42)(28,44)(34,62)(35,52)(37,69)(38,57)(40,71)(41,74)(43,59)(45,58)(47,60)(48,56)(49,51)(50,63)(53,68)(54,73)(55,72)(61,65)(64,70)(66,79)(67,76)(75,78)(77,80);(1,3)(2,6)(4,9)(5,10)(7,12)(8,13)(11,16)(14,15)(17,19)(18,26)(20,34)(21,36)(22,47)(23,37)(24,49)(25,38)(27,50)(28,57)(29,51)(30,52)(31,60)(32,53)(33,46)(35,44)(39,70)(40,64)(41,75)(42,65)(43,67)(45,68)(48,66)(54,79)(55,76)(56,78)(58,69)(59,77)(61,71)(62,74)(63,72)(73,80);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,16)(13,15)(17,20,25,43)(18,27,32,56)(19,34,38,67)(21,40,45,73)(22,41,44,72)(23,48,24,42)(26,50,53,78)(28,54,46,71)(29,55,58,74)(30,59,31,39)(33,61,57,79)(35,63,47,75)(36,64,68,80)(37,66,49,65)(51,76,69,62)(52,77,60,70);(17,21,24,44,25,45,23,22)(18,28,31,29,32,46,30,58)(19,35,37,36,38,47,49,68)(20,40,42,72,43,73,48,41)(26,51,52,57,53,69,60,33)(27,54,39,55,56,71,59,74)(34,63,66,64,67,75,65,80)(50,76,77,79,78,62,70,61)Polycyclic presentation:
2^9|a^2=h,c^2=i,d^2=g,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 393869);