Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6357480 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#127 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1782 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)(3,12)(4,9)(5,10)(6,20)(7,11)(8,14)(13,26)(15,22)(16,24)(17,25)(18,30)(19,23)(21,27)(28,31)(29,32)(33,34)(35,40)(36,48)(37,42)(38,52)(39,43)(41,50)(44,57)(45,54)(46,55)(47,61)(49,56)(51,62)(53,60)(58,64)(59,63);(1,3,6,14,7,15,5,19)(2,8,10,12,11,23,20,22)(4,13,18,27,17,28,16,29)(9,21,24,26,25,32,30,31)(33,35)(34,40)(36,44)(37,46)(38,47)(39,45)(41,53)(42,55)(43,54)(48,57)(49,59)(50,60)(51,58)(52,61)(56,63)(62,64);(1,4,7,17)(2,9,11,25)(3,13,15,28)(5,16,6,18)(8,21,23,32)(10,24,20,30)(12,26,22,31)(14,27,19,29)(33,36,38,50,39,51,37,49)(34,41,42,48,43,56,52,62)(35,44,47,60,45,58,46,59)(40,53,55,57,54,63,61,64);(1,5,7,6)(2,10,11,20)(3,14,15,19)(4,16,17,18)(8,22,23,12)(9,24,25,30)(13,27,28,29)(21,31,32,26)(35,45)(40,54)(44,58)(46,47)(53,63)(55,61)(57,64)(59,60);(4,17)(9,25)(13,28)(16,18)(21,32)(24,30)(26,31)(27,29)(33,37,39,38)(34,42,43,52)(35,46,45,47)(36,49,51,50)(40,55,54,61)(41,48,56,62)(44,59,58,60)(53,57,63,64)Polycyclic presentation:
2^9|b^2=f,c^2=gh,d^2=h,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=dhi,e^c=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264710);