Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33691 |
| 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 | #21365 |
| prime radical | C24 | #137 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
518 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,15,7,14,9,32)(3,16,19,46,20,35,33,42)(4,21,25,54,24,11,26,41)(5,13,6,29,27,44,28,12)(10,30,36,51,38,58,37,18)(17,45,49,63,50,60,57,64)(22,40,31,39,55,43,23,53)(34,59,61,48,52,56,62,47);(1,3,8,19,7,20,9,33)(2,10,15,36,14,38,32,37)(4,17,25,49,24,50,26,57)(5,18,6,30,27,51,28,58)(11,34,41,61,21,52,54,62)(12,35,13,42,29,16,44,46)(22,47,31,59,55,48,23,56)(39,60,43,64,53,45,40,63);(1,4)(2,11)(3,17)(5,22)(6,31)(7,24)(8,25)(9,26)(10,34)(12,39)(13,43)(14,21)(15,41)(16,45)(18,47)(19,49)(20,50)(23,28)(27,55)(29,53)(30,59)(32,54)(33,57)(35,60)(36,61)(37,62)(38,52)(40,44)(42,64)(46,63)(48,51)(56,58);(1,5,8,6,7,27,9,28)(2,12,32,44,14,29,15,13)(3,18,19,30,20,51,33,58)(4,22,25,31,24,55,26,23)(10,35,37,46,38,16,36,42)(11,39,54,40,21,53,41,43)(17,47,49,59,50,48,57,56)(34,60,62,63,52,45,61,64)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=fh,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,h^a=hi,e^b=eh,f^b=fi,h^b=hi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402679);