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 | #2540 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1428 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,6)(3,7,4,8)(9,10)(11,15)(12,13)(14,16)(17,18,22,28)(19,27,32,25)(20,34,37,46)(21,40,33,42)(23,30,24,29)(26,39,45,38)(31,47,48,43)(35,44,36,41);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,32)(18,25,28,27)(20,35,37,36)(21,24,33,23)(26,43,45,47)(29,40,30,42)(31,39,48,38)(34,41,46,44);(10,13)(14,15)(17,20,24,39,22,37,23,38)(18,26,30,46,28,45,29,34)(19,31,21,36,32,48,33,35)(25,41,40,47,27,44,42,43);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,21,24,32,22,33,23,19)(18,27,30,42,28,25,29,40)(20,36,39,48,37,35,38,31)(26,44,46,43,45,41,34,47);(1,4,5,3)(2,7,6,8)(18,28)(20,37)(25,27)(29,30)(31,48)(35,36)(38,39)(40,42)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=h,d^2=fg,e^2=fi,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^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7505514);