Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6357459 |
| 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:
(1,2,7,11)(3,12,16,26)(4,9,18,24)(5,10,6,20)(8,14,22,15)(13,25,28,32)(17,23,19,30)(21,27,31,29)(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);(1,3,6,15,7,16,5,14)(2,8,10,12,11,22,20,26)(4,13,19,29,18,28,17,27)(9,21,23,25,24,31,30,32)(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)(2,9)(3,13)(5,17)(6,19)(7,18)(8,21)(10,23)(11,24)(12,25)(14,27)(15,29)(16,28)(20,30)(22,31)(26,32)(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)(3,14,16,15)(4,17,18,19)(8,12,22,26)(9,23,24,30)(13,27,28,29)(21,25,31,32)(35,45)(40,54)(44,58)(46,47)(53,63)(55,61)(57,64)(59,60);(4,18)(9,24)(13,28)(17,19)(21,31)(23,30)(25,32)(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|a^2=h,b^2=f,c^2=g,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=di,e^c=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264674);