Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #461207 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16383 |
| 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 | |
831 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,8,13,7,14)(3,15,19,9,20,42,18,34)(4,21,24,38,25,10,30,37)(5,26,28,11,29,40,31,41)(16,35,22,32,47,43,46,59)(17,33,50,57,51,61,49,44)(23,54,27,36,56,39,55,52)(45,62,53,60,64,58,48,63);(1,3)(2,9,13,34)(4,22)(5,17)(6,18)(7,19)(8,20)(10,35,21,43)(11,33,41,61)(12,15,14,42)(16,24)(23,53)(25,46)(26,57,40,44)(27,45)(28,49)(29,51)(30,47)(31,50)(32,37,59,38)(36,60,52,63)(39,62,54,58)(48,56)(55,64);(1,4)(2,10)(3,16)(5,27)(6,30)(7,24)(8,25)(9,32)(11,39)(12,38)(13,21)(14,37)(15,43)(17,48)(18,22)(19,46)(20,47)(23,28)(26,52)(29,55)(31,56)(33,60)(34,59)(35,42)(36,40)(41,54)(44,58)(45,49)(50,64)(51,53)(57,62)(61,63);(1,5,6,28,8,29,7,31)(2,11,12,40,13,41,14,26)(3,17,18,49,20,51,19,50)(4,23,24,27,25,56,30,55)(9,33,15,44,34,61,42,57)(10,36,37,39,21,52,38,54)(16,45,46,48,47,64,22,53)(32,58,35,60,59,62,43,63)Polycyclic presentation:
2^9|a^2=fg,b^2=f,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,g^b=gi,d^c=de,e^c=ei,g^c=gi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 368403);