Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394034 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2084 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(5,6,12,18)(7,19,23,37)(8,25,29,14)(9,15,32,43)(10,17,11,16)(13,24,36,22)(20,45,49,38)(21,46,52,60)(26,54,48,34)(27,39,58,53)(28,40,30,41)(31,44,33,42)(35,51,61,50)(47,62,67,65)(55,59,66,68)(56,64,57,63);(1,2)(3,4)(5,7)(6,13)(8,26)(9,21)(10,24)(11,22)(12,23)(14,38)(15,35)(16,37)(17,19)(18,36)(20,30)(25,45)(27,55)(28,49)(29,48)(31,51)(32,52)(33,50)(34,41)(39,62)(40,54)(42,60)(43,61)(44,46)(47,57)(53,65)(56,67)(58,66)(59,64)(63,68);(2,4)(5,8,10,28,12,29,11,30)(6,14,16,40,18,25,17,41)(7,20,22,48,23,49,24,26)(9,27,31,56,32,58,33,57)(13,34,19,45,36,54,37,38)(15,39,42,63,43,53,44,64)(21,47,50,66,52,67,51,55)(35,59,46,65,61,68,60,62);(5,9,12,32)(6,15,18,43)(7,21,23,52)(8,27,29,58)(10,31,11,33)(13,35,36,61)(14,39,25,53)(16,42,17,44)(19,46,37,60)(20,47,49,67)(22,50,24,51)(26,55,48,66)(28,56,30,57)(34,59,54,68)(38,62,45,65)(40,63,41,64)Polycyclic presentation:
2^9|a^2=i,c^2=e,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,f^c=fi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301831);