Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573511 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2401 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1389 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18,22,27)(19,29,32,24)(20,25,34,42)(21,26,35,43)(23,28,37,45)(30,44,36,39)(31,47,48,40)(33,46,38,41);(2,6)(7,8)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,31)(21,36)(22,32)(23,38)(25,40)(26,44)(27,29)(28,46)(30,35)(33,37)(34,48)(39,43)(41,45)(42,47);(1,3,5,4)(2,7,6,8)(10,14)(13,16)(17,20,22,34)(18,25,27,42)(19,30,32,36)(21,37,35,23)(24,39,29,44)(26,45,43,28)(31,33,48,38)(40,41,47,46);(1,4,5,3)(2,8,6,7)(20,21)(23,37)(25,26)(28,45)(30,48)(31,36)(33,38)(34,35)(39,47)(40,44)(41,46)(42,43);(17,21,22,35)(18,26,27,43)(19,31,32,48)(20,23,34,37)(24,40,29,47)(25,28,42,45)(30,38,36,33)(39,46,44,41)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=f,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dh,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6488031);