Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33688 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #3109 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #21580 |
| prime radical | #3054 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
552 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14,6,12,23,15)(3,16,21,38,19,9,34,37)(4,22,26,44,27,40,33,47)(5,11,7,31,29,13,30,45)(10,39,41,25,43,24,42,28)(17,49,52,63,53,60,57,64)(18,46,32,59,54,36,20,55)(35,56,61,58,48,50,62,51);(1,3)(2,9)(4,17)(5,18)(6,19)(7,32)(8,21)(10,35)(11,36)(12,16)(13,46)(14,37)(15,38)(20,30)(22,49)(23,34)(24,50)(25,58)(26,52)(27,53)(28,51)(29,54)(31,55)(33,57)(39,56)(40,60)(41,61)(42,62)(43,48)(44,63)(45,59)(47,64);(1,4,23,26,6,27,8,33)(2,10,14,42,12,43,15,41)(3,17,34,52,19,53,21,57)(5,28,30,39,29,25,7,24)(9,35,37,62,16,48,38,61)(11,44,31,22,13,47,45,40)(18,51,20,56,54,58,32,50)(36,63,55,49,46,64,59,60);(1,5,8,7,6,29,23,30)(2,11,14,31,12,13,15,45)(3,18,21,32,19,54,34,20)(4,24,26,28,27,39,33,25)(9,36,37,55,16,46,38,59)(10,40,41,47,43,22,42,44)(17,50,52,51,53,56,57,58)(35,60,61,64,48,49,62,63)Polycyclic presentation:
2^9|a^2=g,c^2=eh,d^2=g,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,f^b=fi,d^c=dh,e^c=ei,f^c=fh,g^c=gi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403153);