Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #461157 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16381 |
| 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 | |
851 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,13,8,12,7,14)(3,9,19,33,17,31,18,32)(4,10,28,41,22,36,23,37)(5,24,25,11,27,40,26,39)(15,29,20,34,43,56,49,59)(16,44,48,57,46,58,47,30)(21,51,54,35,52,61,53,38)(42,62,63,64,45,60,50,55);(1,3)(2,9,12,31)(4,20,22,49)(5,16)(6,18)(7,19)(8,17)(10,34)(11,30,39,57)(13,32,14,33)(15,23,43,28)(21,50,52,63)(24,44,40,58)(25,47)(26,48)(27,46)(29,41)(35,60)(36,59)(37,56)(38,62)(42,54,45,53)(51,55)(61,64);(1,4)(2,10,12,36)(3,15,17,43)(5,21)(6,23)(7,28)(8,22)(9,29)(11,38,39,35)(13,37,14,41)(16,45,46,42)(18,20,19,49)(24,51,40,61)(25,53)(26,54)(27,52)(30,55)(31,56)(32,34)(33,59)(44,60)(47,50,48,63)(57,64)(58,62);(1,5,8,27)(2,11,12,39)(3,16,17,46)(4,21,22,52)(6,26,7,25)(9,30,31,57)(10,35,36,38)(13,24,14,40)(15,42,43,45)(18,48,19,47)(20,50,49,63)(23,54,28,53)(29,55,56,64)(32,44,33,58)(34,60,59,62)(37,61,41,51)Polycyclic presentation:
2^9|a^2=efg,b^2=f,c^2=e,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,d^c=de,f^c=fi,h^c=hi,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 368354);