Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #368855 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14023 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1650 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,14)(4,20,24,36)(5,25)(6,12)(7,13)(8,17)(9,23,37,54)(10,26)(11,29)(15,30,44,56)(16,31)(18,33)(19,32)(21,40,43,34)(22,50,53,35)(27,39)(28,38)(41,45)(42,59,63,61)(46,57)(47,58)(48,52,60,51)(49,55,62,64);(1,3)(2,8)(4,21,24,43)(5,16)(6,18)(7,19)(9,34,37,40)(10,31)(11,32)(12,14)(13,33)(15,23,44,54)(17,29)(20,30,36,56)(22,49,53,62)(25,41)(26,46)(27,45)(28,47)(35,59,50,61)(38,57)(39,58)(42,52,63,51)(48,55,60,64);(1,4,7,24)(2,9)(3,15)(5,22,28,53)(6,23,29,54)(8,30,33,56)(10,35)(11,36)(12,20)(13,37)(14,40,32,34)(16,42)(17,43)(18,21)(19,44)(25,48)(26,51,27,52)(31,55,58,64)(38,60)(39,50)(41,61,57,59)(45,49)(46,62)(47,63);(1,5,7,28)(2,10,13,39)(3,16,19,47)(4,22,24,53)(6,27,29,26)(8,31,33,58)(9,35,37,50)(11,38,12,25)(14,41,32,57)(15,42,44,63)(17,46,18,45)(20,48,36,60)(21,49,43,62)(23,52,54,51)(30,55,56,64)(34,59,40,61)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=fg,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^c=hi,e^d=ei77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 276052);