Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 | #45236 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1711 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)(3,8)(4,6)(5,7)(9,10)(11,15)(12,14)(13,16)(17,18)(19,23)(20,24)(21,26)(22,25)(27,34)(28,35)(29,37)(30,36)(31,39)(32,38)(33,40)(41,45)(42,47)(43,46)(44,48);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,19,22,30)(18,23,25,36)(20,27,31,42)(21,28,33,29)(24,34,39,47)(26,35,40,37)(32,41,44,43)(38,45,48,46);(10,14)(11,16)(17,20,22,31)(18,24,25,39)(19,27,30,42)(21,32,33,44)(23,34,36,47)(26,38,40,48)(28,41,29,43)(35,45,37,46);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(19,28)(21,33)(23,35)(26,40)(27,41)(29,30)(32,44)(34,45)(36,37)(38,48)(42,43)(46,47);(2,7)(6,8)(10,14)(13,15)(18,25)(20,31)(23,36)(26,40)(27,42)(32,44)(35,37)(41,43)Polycyclic presentation:
2^9|b^2=fi,c^2=i,d^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,e^c=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7456167);