Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #384334 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14377 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
977 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,12,6,29)(3,14,18,39,19,9,17,33)(4,20,25,35,24,50,23,10)(5,26,27,37,28,38,30,11)(15,41,44,31,21,51,43,40)(16,45,46,32,47,56,48,36)(22,34,53,57,54,60,55,49)(42,59,62,64,52,58,63,61);(1,3,8,19)(2,9,12,14)(4,21,24,15)(5,16,28,47)(6,18,7,17)(10,31,35,40)(11,36,37,32)(13,39,29,33)(20,41,50,51)(22,52,54,42)(23,43,25,44)(26,56,38,45)(27,48,30,46)(34,59,60,58)(49,64,57,61)(53,62,55,63);(1,4,6,23,8,24,7,25)(2,10,13,20,12,35,29,50)(3,15,17,43,19,21,18,44)(5,22,27,53,28,54,30,55)(9,31,33,51,14,40,39,41)(11,34,38,49,37,60,26,57)(16,42,46,62,47,52,48,63)(32,58,45,64,36,59,56,61);(1,5)(2,11,12,37)(3,16)(4,22)(6,27)(7,30)(8,28)(9,32,14,36)(10,34,35,60)(13,38,29,26)(15,42)(17,46)(18,48)(19,47)(20,49,50,57)(21,52)(23,53)(24,54)(25,55)(31,58,40,59)(33,45,39,56)(41,61,51,64)(43,62)(44,63)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=f,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,g^c=gi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291533);