Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #198135 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#121 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
973 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,10,7,12,5,11)(3,13,16,23,17,26,15,8)(4,18,21,9,19,25,20,24)(14,22,30,32,28,31,29,27)(33,34)(35,45,49,55)(36,50,53,57)(37,43)(38,42)(39,44)(40,48,56,47)(41,51,58,52)(46,60)(54,63)(59,61)(62,64);(1,3,7,17)(2,8)(4,14,19,28)(5,15,6,16)(9,22)(10,13)(11,26)(12,23)(18,27)(20,29,21,30)(24,31)(25,32)(33,35,37,47,39,49,38,48)(34,40,43,55,44,56,42,45)(36,46,51,61,53,63,52,62)(41,54,50,64,58,60,57,59);(1,4)(2,9)(3,14)(5,21)(6,20)(7,19)(8,22)(10,18)(11,25)(12,24)(13,27)(15,30)(16,29)(17,28)(23,31)(26,32)(33,36)(34,41)(35,46)(37,51)(38,52)(39,53)(40,54)(42,57)(43,50)(44,58)(45,59)(47,61)(48,62)(49,63)(55,64)(56,60);(4,19)(9,24)(14,28)(18,25)(20,21)(22,31)(27,32)(29,30)(33,37,39,38)(34,42,44,43)(35,47,49,48)(36,51,53,52)(40,45,56,55)(41,57,58,50)(46,61,63,62)(54,59,60,64)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=i,e^2=hi,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=ei,f^b=fhi,g^b=gi,d^c=dh,e^c=ehi,f^c=fhi,g^c=ghi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 105409);