Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #187676 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #62828 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2050 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,14,9,3,16,21)(4,23,27,58,29,52,48,57)(5,11,32,13,34,18,49,20)(7,37,35,30,39,54,33,45)(8,15,36,12,31,22,50,19)(10,40,43,63,17,24,42,26)(25,41,28,47,62,51,59,56)(38,44,64,46,60,53,61,55);(1,3)(2,9)(4,17)(5,30)(6,12)(7,37)(8,22)(10,29)(11,35)(13,32)(14,36)(15,31)(16,19)(18,33)(20,49)(21,50)(23,38)(24,58)(25,42)(26,47)(27,46)(28,53)(34,45)(39,54)(40,57)(41,64)(43,62)(44,59)(48,55)(51,61)(52,60)(56,63);(1,4)(2,10)(3,17)(5,24)(6,27)(7,38)(8,28)(9,29)(11,41)(12,46)(13,47)(14,43)(15,44)(16,48)(18,51)(19,55)(20,56)(21,42)(22,53)(23,37)(25,50)(26,32)(30,58)(31,59)(33,61)(34,40)(35,64)(36,62)(39,60)(45,57)(49,63)(52,54);(1,5,31,35)(2,11,22,30)(3,18,15,45)(4,24,59,64)(6,32,50,39)(7,16,49,36)(8,33,9,34)(10,41,53,58)(12,37,21,20)(13,19,54,14)(17,51,44,57)(23,42,56,46)(25,60,27,26)(28,61,29,40)(38,48,63,62)(43,47,55,52)Polycyclic presentation:
2^9|a^2=g,d^2=hi,e^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dh,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,f^c=fh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94938);