Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18346 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C2×#121 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
985 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,11,7,12,5,10)(3,13,16,8,17,23,15,24)(4,18,19,9,21,25,20,26)(14,22,28,31,30,32,29,27)(33,34)(35,45,49,55)(36,50,53,57)(37,43)(38,42)(39,44)(40,47,56,48)(41,52,58,51)(46,60)(54,63)(59,61)(62,64);(1,3)(2,8)(4,14)(5,16)(6,15)(7,17)(9,22)(10,23)(11,13)(12,24)(18,31)(19,29)(20,28)(21,30)(25,27)(26,32)(33,35,37,48,39,49,38,47)(34,40,43,55,44,56,42,45)(36,46,52,61,53,63,51,62)(41,54,50,64,58,60,57,59);(1,4)(2,9)(3,14)(5,20)(6,19)(7,21)(8,22)(10,18)(11,25)(12,26)(13,27)(15,29)(16,28)(17,30)(23,31)(24,32)(33,36)(34,41)(35,46)(37,52)(38,51)(39,53)(40,54)(42,57)(43,50)(44,58)(45,59)(47,62)(48,61)(49,63)(55,64)(56,60);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,19,21,20)(8,13,24,23)(9,25,26,18)(14,28,30,29)(22,31,32,27)(33,37,39,38)(34,42,44,43)(35,47,49,48)(36,51,53,52)(40,55,56,45)(41,50,58,57)(46,61,63,62)(54,59,60,64)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=h,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=di,f^a=fhi,d^b=dh,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,f^c=fh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 102661);