Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19552 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| socle | C22 | C22×#121 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1514 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)(3,12,14,26)(4,9,17,23)(5,10,6,20)(8,15,22,16)(13,25,29,32)(18,24,19,30)(21,27,31,28)(33,34,39,44)(35,45,49,55)(36,50,51,58)(37,43,38,42)(40,48,56,47)(41,53,57,52)(46,60,61,54)(59,63,64,62);(1,3,5,15,7,14,6,16)(2,8,20,26,11,22,10,12)(4,13,18,27,17,29,19,28)(9,21,30,32,23,31,24,25)(33,35,38,48,39,49,37,47)(34,40,43,55,44,56,42,45)(36,46,53,63,51,61,52,62)(41,54,58,64,57,60,50,59);(1,4)(2,9)(3,13)(5,18)(6,19)(7,17)(8,21)(10,24)(11,23)(12,25)(14,29)(15,27)(16,28)(20,30)(22,31)(26,32)(33,36)(34,41)(35,46)(37,52)(38,53)(39,51)(40,54)(42,50)(43,58)(44,57)(45,59)(47,62)(48,63)(49,61)(55,64)(56,60);(3,14)(4,17)(8,22)(9,23)(12,26)(15,16)(18,19)(24,30)(36,51)(41,57)(46,61)(50,58)(52,53)(54,60)(59,64)(62,63);(1,5,7,6)(2,10,11,20)(3,15,14,16)(4,18,17,19)(8,12,22,26)(9,24,23,30)(13,27,29,28)(21,25,31,32)Polycyclic presentation:
2^9|a^2=h,b^2=fh,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=dhi,d^c=dh,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6261899);