Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #198534 |
| 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 | |
984 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)(3,13,16,44)(4,18)(5,22)(10,36,38,28)(12,30,29,56)(14,45,48,42)(15,43,24,17)(19,52,39,53)(20,37)(21,51)(26,41)(27,40)(31,59)(32,33)(46,62)(47,49,54,64)(50,55)(57,61)(60,63);(1,2,5,23,25,29,41,30)(3,9,24,32,43,58,13,40)(4,19,20,52,51,10,37,36)(6,12,22,56,7,11,26,8)(14,46,54,63,64,31,42,60)(15,27,17,34,44,33,16,35)(18,39,50,53,21,38,55,28)(45,57,48,62,47,61,49,59);(1,3)(2,9)(4,14)(5,24)(6,17)(7,16)(8,27)(10,31)(11,35)(12,34)(13,41)(15,26)(18,48)(19,46)(20,54)(21,49)(22,44)(23,32)(25,43)(28,57)(29,58)(30,40)(33,56)(36,60)(37,42)(38,59)(39,62)(45,55)(47,50)(51,64)(52,63)(53,61);(1,4,6,21)(2,10,12,39)(3,14,17,49)(5,20,22,55)(7,18,25,51)(8,28,30,52)(9,31,34,62)(11,38,29,19)(13,42,15,47)(16,48,43,64)(23,36,56,53)(24,54,44,45)(26,50,41,37)(27,57,40,63)(32,60,33,61)(35,59,58,46)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=dhi,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, 105731);