Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6356405 |
| 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 | #537 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1441 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)(3,12)(4,9)(5,10)(6,20)(7,11)(8,14)(13,26)(15,22)(16,24)(17,25)(18,30)(19,23)(21,28)(27,32)(29,31)(33,34,38,42)(35,43,46,56)(36,40,47,54)(37,49,51,41)(39,50,53,45)(44,57,58,64)(48,61,60,55)(52,62,63,59);(1,3,6,14,7,15,5,19)(2,8,10,12,11,23,20,22)(4,13,18,28,16,27,17,29)(9,21,25,26,24,31,30,32);(1,4)(2,9)(3,13)(5,17)(6,18)(7,16)(8,21)(10,25)(11,24)(12,26)(14,28)(15,27)(19,29)(20,30)(22,32)(23,31)(33,35,38,46)(34,39,42,53)(36,44,47,58)(37,50,51,45)(40,52,54,63)(41,56,49,43)(48,62,60,59)(55,64,61,57);(4,16)(9,24)(13,27)(17,18)(21,31)(25,30)(26,32)(28,29)(33,36)(34,40)(35,44)(37,48)(38,47)(39,52)(41,55)(42,54)(43,57)(45,59)(46,58)(49,61)(50,62)(51,60)(53,63)(56,64);(1,5,7,6)(2,10,11,20)(3,14,15,19)(4,17,16,18)(8,22,23,12)(9,25,24,30)(13,28,27,29)(21,32,31,26)(36,47)(40,54)(44,58)(48,60)(52,63)(55,61)(57,64)(59,62)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,e^b=eh,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6263642);