Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18446 |
| 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 | #3444 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
975 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)(4,18,21,9,19,26,20,25)(14,22,30,31,28,32,29,27)(33,34)(35,45,49,56)(36,50,51,57)(37,43)(38,44)(39,42)(40,47,55,48)(41,53,58,52)(46,60)(54,61)(59,62)(63,64);(1,3)(2,8)(4,14)(5,16)(6,15)(7,17)(9,22)(10,23)(11,13)(12,24)(18,31)(19,28)(20,30)(21,29)(25,32)(26,27)(33,35,37,48,39,49,38,47)(34,40,43,56,42,55,44,45)(36,46,53,62,51,61,52,63)(41,54,50,64,58,60,57,59);(1,4)(2,9)(3,14)(5,20)(6,21)(7,19)(8,22)(10,18)(11,26)(12,25)(13,27)(15,29)(16,30)(17,28)(23,31)(24,32)(33,36)(34,41)(35,46)(37,53)(38,52)(39,51)(40,54)(42,58)(43,50)(44,57)(45,59)(47,63)(48,62)(49,61)(55,60)(56,64);(4,19)(9,25)(14,28)(18,26)(20,21)(22,32)(27,31)(29,30)(34,42)(36,51)(40,55)(43,44)(45,56)(46,61)(52,53)(62,63)Polycyclic presentation:
2^9|a^2=g,b^2=eg,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=di,f^a=fhi,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, 102654);