Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #479799 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17255 |
| 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 | |
730 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)(3,13)(4,19,22,42)(5,10,27,40)(6,11)(7,12)(8,17)(9,23,36,29)(14,44)(15,46,20,52)(16,45,49,32)(18,41)(21,55)(24,59,50,37)(25,53)(26,39,30,43)(28,33)(31,51,34,48)(35,58)(38,57)(47,63)(54,64)(56,62)(60,61);(1,3,14,18,7,17,6,28)(2,8,11,41,12,13,44,33)(4,20,29,51,22,15,23,48)(5,24,30,16,27,50,26,49)(9,34,19,52,36,31,42,46)(10,37,39,45,40,59,43,32)(21,56,58,47,25,60,57,54)(35,63,55,62,38,64,53,61);(1,4,6,29,7,22,14,23)(2,9,11,42,12,36,44,19)(3,15,28,48,17,20,18,51)(5,25,26,57,27,21,30,58)(8,31,41,52,13,34,33,46)(10,38,39,55,40,35,43,53)(16,47,24,60,49,54,50,56)(32,61,37,64,45,62,59,63);(1,5)(2,10)(3,16)(4,21)(6,30)(7,27)(8,32)(9,35)(11,43)(12,40)(13,45)(14,26)(15,47)(17,49)(18,24)(19,53)(20,54)(22,25)(23,58)(28,50)(29,57)(31,61)(33,37)(34,62)(36,38)(39,44)(41,59)(42,55)(46,64)(48,56)(51,60)(52,63)Polycyclic presentation:
2^9|a^2=eg,b^2=efh,c^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=def,e^b=ei,f^b=fi,g^b=gi,d^c=de,e^c=ei,f^c=fi,h^c=hi,f^d=fi,g^d=gi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386995);