Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #461255 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16384 |
| 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 | |
793 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)(3,14,19,43)(4,20)(5,26)(6,12)(7,31)(8,13)(9,17,34,18)(10,24)(11,28)(15,32,21,52)(16,33,50,62)(22,54)(23,36)(25,37)(27,58)(29,40)(30,39)(35,56)(38,55)(41,45,51,46)(42,49,57,48)(44,61,53,63)(47,64,59,60);(1,3,7,18,8,19,6,17)(2,9,12,14,13,34,31,43)(4,21,24,45,25,15,23,46)(5,16,29,49,30,50,28,48)(10,32,20,51,36,52,37,41)(11,33,39,42,40,62,26,57)(22,53,55,47,56,44,27,59)(35,60,38,61,54,64,58,63);(1,4)(2,10)(3,15)(5,27)(6,24)(7,23)(8,25)(9,32)(11,38)(12,37)(13,36)(14,41)(16,47)(17,46)(18,45)(19,21)(20,31)(22,28)(26,54)(29,56)(30,55)(33,61)(34,52)(35,39)(40,58)(42,60)(43,51)(44,48)(49,53)(50,59)(57,64)(62,63);(1,5,6,28,8,30,7,29)(2,11,12,39,13,40,31,26)(3,16,17,48,19,50,18,49)(4,22,23,27,25,56,24,55)(9,33,14,42,34,62,43,57)(10,35,20,38,36,54,37,58)(15,44,45,47,21,53,46,59)(32,60,51,61,52,64,41,63)Polycyclic presentation:
2^9|a^2=efg,b^2=f,d^2=e,e^2=i,f^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,f^c=fi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 368451);