Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #421443 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15291 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
903 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)(3,15,20,34)(4,21,25,42)(5,26,30,11)(6,13,7,12)(9,19,35,18)(10,31,38,24)(16,44,47,32)(17,39,49,55)(22,36,46,43)(23,52,54,50)(27,33,48,60)(28,40,29,41)(37,53,56,58)(45,63,64,59)(51,61,57,62);(1,3)(2,9)(4,22,25,46)(5,27)(6,19)(7,18)(8,20)(10,36,38,43)(11,39)(12,34)(13,15)(14,35)(16,24,47,31)(17,28)(21,44,42,32)(23,45,54,64)(26,55)(29,49)(30,48)(33,40)(37,59,56,63)(41,60)(50,61,52,62)(51,58,57,53);(1,4)(2,10)(3,16)(5,23)(6,24)(7,31)(8,25)(9,32)(11,37)(12,21)(13,42)(14,38)(15,43)(17,45)(18,46)(19,22)(20,47)(26,56)(27,57)(28,53)(29,58)(30,54)(33,59)(34,36)(35,44)(39,62)(40,50)(41,52)(48,51)(49,64)(55,61)(60,63);(1,5,6,28,8,30,7,29)(2,11,12,40,14,26,13,41)(3,17,18,27,20,49,19,48)(4,23,24,53,25,54,31,58)(9,33,15,39,35,60,34,55)(10,37,21,50,38,56,42,52)(16,45,46,57,47,64,22,51)(32,59,43,62,44,63,36,61)Polycyclic presentation:
2^9|a^2=i,b^2=ef,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,f^c=fi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 328640);