Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6351720 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2595 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)(6,12)(7,11)(8,16)(9,19)(14,22)(15,23)(17,26)(20,24)(21,25)(27,28)(29,31)(30,32)(33,34)(35,45)(36,41)(37,44)(38,42)(39,43)(40,47)(46,60)(48,57)(49,58)(50,56)(51,54)(52,55)(53,62)(59,63)(61,64);(1,3)(2,8)(4,14)(5,15)(6,16)(7,17)(9,22)(10,13)(11,23)(12,26)(18,31)(19,28)(20,30)(21,29)(24,27)(25,32)(33,35,39,48)(34,40,43,54)(36,46,50,63)(37,47,38,51)(41,53,56,64)(42,45,44,57)(49,61,52,62)(55,59,58,60);(1,4)(2,9)(3,14)(5,19)(6,21)(7,20)(8,22)(10,24)(11,25)(12,18)(13,27)(15,28)(16,29)(17,30)(23,32)(26,31)(33,36)(34,41)(35,46)(37,49)(38,52)(39,50)(40,53)(42,55)(43,56)(44,58)(45,59)(47,61)(48,63)(51,62)(54,64)(57,60);(1,5,7,6)(2,10,11,12)(3,15,17,16)(4,19,20,21)(8,13,23,26)(9,24,25,18)(14,28,30,29)(22,27,32,31)(33,37,39,38)(34,42,43,44)(35,47,48,51)(36,49,50,52)(40,45,54,57)(41,55,56,58)(46,61,63,62)(53,59,64,60);(2,11)(8,23)(9,25)(10,12)(13,26)(18,24)(22,32)(27,31)(34,43)(40,54)(41,56)(42,44)(45,57)(53,64)(55,58)(59,60)Polycyclic presentation:
2^9|b^2=i,d^2=h,f^2=h,g^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=ghi,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, 6258923);