Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6629034 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3067 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2922 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 68 points:
(2,4)(5,6,12,18)(7,19,24,35)(8,14,26,38)(9,29,31,42)(10,17,11,16)(13,23,36,22)(15,32,41,30)(20,43,46,59)(21,49,51,34)(25,53,54,62)(27,40,28,39)(33,48,58,47)(37,56,61,55)(44,52,60,50)(45,64,65,57)(63,67,68,66);(5,7,11,23,12,24,10,22)(6,13,17,35,18,36,16,19)(8,20,28,48,26,46,27,47)(9,21,32,52,31,51,30,50)(14,33,40,59,38,58,39,43)(15,34,42,60,41,49,29,44)(25,45,56,67,54,65,55,66)(37,57,62,68,61,64,53,63);(1,2,3,4)(5,8)(6,14)(7,20)(9,25)(10,27)(11,28)(12,26)(13,33)(15,37)(16,39)(17,40)(18,38)(19,43)(21,45)(22,47)(23,48)(24,46)(29,53)(30,55)(31,54)(32,56)(34,57)(35,59)(36,58)(41,61)(42,62)(44,63)(49,64)(50,66)(51,65)(52,67)(60,68);(5,9)(6,15)(7,21)(8,25)(10,30)(11,32)(12,31)(13,34)(14,37)(16,29)(17,42)(18,41)(19,44)(20,45)(22,50)(23,52)(24,51)(26,54)(27,55)(28,56)(33,57)(35,60)(36,49)(38,61)(39,53)(40,62)(43,63)(46,65)(47,66)(48,67)(58,64)(59,68);(8,26)(14,38)(20,46)(25,54)(27,28)(33,58)(37,61)(39,40)(43,59)(45,65)(47,48)(53,62)(55,56)(57,64)(63,68)(66,67)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,h^a=hi,e^c=ei,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6537279);