Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6351787 |
| 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 | |
2623 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)(4,17)(5,12)(6,11)(7,10)(8,15)(9,19)(14,22)(16,25)(18,26)(20,23)(21,24)(27,30)(28,32)(29,31)(33,34)(35,45)(36,41)(37,44)(38,43)(39,42)(40,47)(46,60)(48,57)(49,56)(50,58)(51,54)(52,55)(53,63)(59,61)(62,64);(1,3)(2,8)(4,14)(5,20)(6,15)(7,16)(9,22)(10,23)(11,25)(12,13)(17,31)(18,28)(19,30)(21,29)(24,32)(26,27)(33,35)(34,40)(36,46)(37,47)(38,51)(39,48)(41,53)(42,54)(43,45)(44,57)(49,61)(50,62)(52,63)(55,59)(56,64)(58,60);(1,4)(2,9)(3,14)(5,19)(6,21)(7,18)(8,22)(10,24)(11,17)(12,26)(13,27)(15,29)(16,28)(20,30)(23,32)(25,31)(33,36)(34,41)(35,46)(37,50)(38,52)(39,49)(40,53)(42,56)(43,55)(44,58)(45,59)(47,62)(48,61)(51,63)(54,64)(57,60);(4,18)(9,24)(14,28)(17,26)(19,21)(22,32)(27,31)(29,30)(34,42)(36,49)(40,54)(43,44)(45,57)(46,61)(50,52)(62,63);(2,10)(8,23)(9,24)(11,12)(13,25)(17,26)(22,32)(27,31)(33,37,39,38)(34,43,42,44)(35,47,48,51)(36,50,49,52)(40,45,54,57)(41,55,56,58)(46,62,61,63)(53,59,64,60)Polycyclic presentation:
2^9|e^2=i,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,d^c=dh,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, 6259004);