Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518920 |
| 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×#459 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1366 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)(19,25)(20,34)(21,39)(22,28)(23,29)(24,30)(26,38)(27,32)(31,43)(33,42)(35,41)(36,44)(37,46)(40,45)(47,48);(2,6)(7,8)(11,16)(13,15)(17,19,23,33)(18,25,29,42)(20,35,38,31)(21,22,32,24)(26,43,34,41)(27,30,39,28)(36,37,47,40)(44,46,48,45);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,20,23,38)(18,26,29,34)(19,31,33,35)(21,36,32,47)(22,40,24,37)(25,41,42,43)(27,44,39,48)(28,46,30,45);(10,14)(13,15)(17,21,24,19,23,32,22,33)(18,27,30,42,29,39,28,25)(20,36,37,31,38,47,40,35)(26,44,45,43,34,48,46,41);(1,4,5,3)(2,7,6,8)(17,22,23,24)(18,28,29,30)(19,32,33,21)(20,37,38,40)(25,27,42,39)(26,45,34,46)(31,36,35,47)(41,48,43,44)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=fg,d^2=h,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7522726);