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 | |
2597 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)(35,43)(36,40)(37,49)(38,41)(39,45)(42,52)(44,60)(46,57)(47,55)(48,58)(50,54)(51,56)(53,62)(59,61)(63,64);(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)(33,35,38,46)(34,39,41,54)(36,44,47,61)(37,50,52,45)(40,53,55,64)(42,57,49,43)(48,63,51,62)(56,59,58,60);(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)(34,40)(35,44)(37,51)(38,47)(39,53)(41,55)(42,58)(43,59)(45,63)(46,61)(48,52)(49,56)(50,62)(54,64)(57,60);(4,18)(9,24)(14,28)(17,26)(19,21)(22,32)(27,31)(29,30)(34,41)(36,47)(39,54)(42,49)(43,57)(44,61)(48,51)(62,63);(2,10)(8,23)(9,24)(11,12)(13,25)(17,26)(22,32)(27,31)(34,41)(39,54)(40,55)(42,49)(43,57)(53,64)(56,58)(59,60)Polycyclic presentation:
2^9|b^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, 6258989);