Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #198510 |
| 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 | C23×C4 | C24 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
992 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(2,8,11,23)(3,13,16,43)(4,18)(5,22)(10,36,38,28)(12,30,29,56)(14,44,47,41)(15,42,24,17)(19,52,49,53)(20,54)(21,51)(26,40)(27,39)(31,59)(32,33)(37,50)(45,64)(46,48,55,63)(57,61)(60,62);(1,2,5,23,25,29,40,30)(3,9,24,32,42,58,13,39)(4,19,20,52,51,38,54,28)(6,12,22,56,7,11,26,8)(10,37,36,18,49,50,53,21)(14,45,55,61,63,59,41,57)(15,27,17,34,43,33,16,35)(31,44,60,47,64,46,62,48);(1,3)(2,9)(4,14)(5,24)(6,17)(7,16)(8,27)(10,31)(11,35)(12,34)(13,40)(15,26)(18,47)(19,45)(20,55)(21,48)(22,43)(23,32)(25,42)(28,57)(29,58)(30,39)(33,56)(36,60)(37,44)(38,59)(41,54)(46,50)(49,64)(51,63)(52,61)(53,62);(1,4,6,21)(2,10,12,19)(3,14,17,48)(5,20,22,37)(7,18,25,51)(8,28,30,53)(9,31,34,45)(11,38,29,49)(13,41,15,46)(16,47,42,63)(23,36,56,52)(24,55,43,44)(26,50,40,54)(27,57,39,62)(32,60,33,61)(35,59,58,64)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=h,e^2=hi,f^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,d^b=dh,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fhi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 105707);