Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1505 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)(9,10,12,14)(11,15,16,13)(17,18)(19,24)(20,25)(21,35)(22,27)(23,28)(26,30)(29,37)(31,38)(32,45)(33,40)(34,41)(36,42)(39,44)(43,47)(46,48);(2,6)(7,8)(11,16)(13,15)(17,19,23,31)(18,24,28,38)(20,29,33,43)(21,36,30,22)(25,37,40,47)(26,27,35,42)(32,46,44,34)(39,41,45,48);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,20,23,33)(18,25,28,40)(19,29,31,43)(21,32,30,44)(22,34,36,46)(24,37,38,47)(26,39,35,45)(27,41,42,48);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(17,21,22,19,23,30,36,31)(18,26,27,38,28,35,42,24)(20,32,34,29,33,44,46,43)(25,39,41,47,40,45,48,37);(2,6)(7,8)(20,33)(25,40)(29,43)(32,44)(34,46)(37,47)(39,45)(41,48)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=fgi,d^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7522235);