Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19086 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| socle | C22 | C22×#120 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2588 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,13)(4,18)(5,10)(6,11)(7,12)(8,17)(9,19)(14,22)(15,26)(16,23)(20,24)(21,25)(27,29)(28,32)(30,31)(33,34)(35,44)(36,41)(37,43)(38,50)(39,42)(40,47)(45,60)(46,54)(48,57)(49,56)(51,55)(52,58)(53,63)(59,61)(62,64);(1,3)(2,8)(4,14)(5,16)(6,17)(7,15)(9,22)(10,13)(11,26)(12,23)(18,31)(19,29)(20,28)(21,30)(24,27)(25,32)(33,35,39,46)(34,40,42,55)(36,45,49,61)(37,47,38,51)(41,53,56,64)(43,54,50,44)(48,62,52,63)(57,60,58,59);(1,4)(2,9)(3,14)(5,19)(6,21)(7,20)(8,22)(10,24)(11,18)(12,25)(13,27)(15,28)(16,29)(17,30)(23,32)(26,31)(33,36)(34,41)(35,45)(37,48)(38,52)(39,49)(40,53)(42,56)(43,57)(44,59)(46,61)(47,62)(50,58)(51,63)(54,60)(55,64);(3,15)(8,23)(13,26)(14,28)(16,17)(22,32)(27,31)(29,30)(34,42)(35,46)(41,56)(43,50)(45,61)(47,51)(57,58)(62,63);(1,5,7,6)(2,10,12,11)(3,16,15,17)(4,19,20,21)(8,13,23,26)(9,24,25,18)(14,29,28,30)(22,27,32,31)(33,37,39,38)(34,43,42,50)(35,47,46,51)(36,48,49,52)(40,54,55,44)(41,57,56,58)(45,62,61,63)(53,60,64,59)Polycyclic presentation:
2^9|b^2=i,e^2=h,f^2=h,g^2=hi,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=ghi,d^b=dh,f^b=fh,g^b=gi,f^c=fi,g^c=ghi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6258909);