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×#550 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1497 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)(11,15)(12,13)(14,16)(17,18)(19,30)(20,34)(21,39)(22,29)(23,28)(24,27)(25,32)(26,38)(31,46)(33,41)(35,45)(36,44)(37,43)(40,48)(42,47);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,32)(18,25,28,30)(20,35,37,48)(21,24,33,22)(26,42,44,46)(27,39,29,41)(31,36,47,38)(34,40,43,45);(10,13)(14,15)(17,20,24,38,23,37,22,36)(18,26,29,34,28,44,27,43)(19,31,21,35,32,47,33,48)(25,40,39,42,30,45,41,46);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,21,22,19,23,33,24,32)(18,25,27,41,28,30,29,39)(20,35,36,31,37,48,38,47)(26,40,43,46,44,45,34,42);(1,4,5,3)(2,7,6,8)(17,22,23,24)(18,27,28,29)(19,21,32,33)(20,36,37,38)(25,39,30,41)(26,43,44,34)(31,35,47,48)(40,42,45,46)Polycyclic presentation:
2^9|b^2=i,c^2=hi,d^2=fg,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7505159);