Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #383887 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14365 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1475 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)(3,8,15,33)(4,17)(5,22,25,38)(6,26)(7,11)(9,20)(10,24,37,28)(12,29)(13,40,18,49)(14,43)(16,46,45,32)(19,52,54,35)(21,39)(23,56)(27,36)(30,47,34,42)(31,44)(41,58)(48,59)(50,53,57,55)(51,60)(61,62)(63,64);(1,3,6,16,7,15,29,45)(2,8,12,32,11,33,26,46)(4,18,20,47,21,13,27,42)(5,23,24,48,25,14,28,44)(9,34,17,49,36,30,39,40)(10,31,38,43,37,59,22,56)(19,41,53,62,54,51,55,63)(35,60,57,61,52,58,50,64);(1,4,7,21)(2,9,11,36)(3,13,15,18)(5,19,25,54)(6,27,29,20)(8,30,33,34)(10,35,37,52)(12,39,26,17)(14,41,23,51)(16,47,45,42)(22,57,38,50)(24,55,28,53)(31,58,59,60)(32,49,46,40)(43,61,56,64)(44,63,48,62);(1,5)(2,10)(3,14)(4,19)(6,28)(7,25)(8,31)(9,35)(11,37)(12,22)(13,41)(15,23)(16,48)(17,50)(18,51)(20,55)(21,54)(24,29)(26,38)(27,53)(30,58)(32,56)(33,59)(34,60)(36,52)(39,57)(40,61)(42,63)(43,46)(44,45)(47,62)(49,64)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=i,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,f^c=fi,f^d=fi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291084);