Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #488723 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #12164 | C2 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2960 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
678 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 678 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:
(4,11)(6,12)(7,14)(9,15)(17,18)(19,33)(20,40)(21,29)(22,48)(23,31)(24,30)(25,32)(26,36)(27,43)(28,46)(34,67)(35,55)(37,53)(38,66)(39,60)(41,61)(42,63)(44,64)(45,57)(47,62)(49,54)(50,58)(51,59)(52,68)(56,73)(65,70)(69,76)(71,77)(72,80)(74,79)(75,78);(1,2)(3,6)(4,7)(5,8)(9,13)(10,12)(11,14)(15,16)(17,19)(18,26)(20,34)(21,36)(22,49)(23,38)(24,37)(25,39)(27,52)(28,60)(29,53)(30,55)(31,54)(32,56)(33,48)(35,46)(40,76)(41,69)(42,72)(43,75)(44,71)(45,74)(47,73)(50,68)(51,70)(57,65)(58,77)(59,80)(61,78)(62,66)(63,67)(64,79);(1,3,5,10)(2,6,8,12)(4,9,11,15)(7,13,14,16)(17,20,25,44)(18,27,32,59)(19,34,39,71)(21,45,47,41)(22,50,46,42)(23,51,24,43)(26,52,56,80)(28,61,48,57)(29,63,62,58)(30,64,31,40)(33,65,60,78)(35,72,49,68)(36,74,73,69)(37,75,38,70)(53,67,66,77)(54,76,55,79);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)(17,21,24,46,25,47,23,22)(18,28,31,29,32,48,30,62)(19,35,38,36,39,49,37,73)(20,41,43,50,44,45,51,42)(26,53,55,60,56,66,54,33)(27,57,40,58,59,61,64,63)(34,68,70,69,71,72,75,74)(52,77,79,65,80,67,76,78)Polycyclic presentation:
2^9|c^2=h,d^2=ghi,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dghi,e^a=eg,f^a=fi,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,d^c=di,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, 396027);