Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6356901 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #3482 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1447 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,6,10)(3,11,13,26)(4,8,17,24)(5,19,12,9)(7,20,21,15)(14,27,16,30)(18,32,28,25)(22,31,23,29)(33,34)(35,45)(36,50)(37,42)(38,43)(39,44)(40,48)(41,52)(46,58)(47,57)(49,56)(51,55)(53,59)(54,60)(61,62)(63,64);(1,3,12,20,6,13,5,15)(2,7,19,26,10,21,9,11)(4,16,28,29,17,14,18,31)(8,23,32,27,24,22,25,30)(33,35,38,49,39,47,37,48)(34,40,42,57,44,56,43,45)(36,51,54,62,53,46,52,63)(41,58,59,61,60,55,50,64);(3,13)(7,21)(11,26)(14,16)(15,20)(22,23)(27,30)(29,31)(33,36,39,53)(34,41,44,60)(35,46,47,51)(37,52,38,54)(40,55,56,58)(42,59,43,50)(45,61,57,64)(48,62,49,63);(1,4)(2,8)(3,14)(5,18)(6,17)(7,22)(9,25)(10,24)(11,27)(12,28)(13,16)(15,29)(19,32)(20,31)(21,23)(26,30)(35,47)(40,56)(45,57)(46,51)(48,49)(55,58)(61,64)(62,63);(4,17)(8,24)(14,16)(18,28)(22,23)(25,32)(27,30)(29,31)(33,37,39,38)(34,42,44,43)(35,48,47,49)(36,52,53,54)(40,57,56,45)(41,59,60,50)(46,62,51,63)(55,64,58,61)Polycyclic presentation:
2^9|a^2=hi,b^2=fh,c^2=i,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ch,d^b=dh,g^c=gi,e^d=ehi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264181);