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, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1321 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,15)(12,13)(14,16)(17,18,23,29)(19,27,32,25)(20,34,37,45)(21,39,33,42)(22,30,24,28)(26,38,44,36)(31,47,48,43)(35,46,40,41);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,25)(20,35)(21,22)(23,32)(24,33)(26,43)(27,29)(28,39)(30,42)(31,36)(34,41)(37,40)(38,48)(44,47)(45,46);(10,13)(14,15)(17,20,24,38,23,37,22,36)(18,26,30,45,29,44,28,34)(19,31,21,40,32,48,33,35)(25,41,39,47,27,46,42,43);(9,11,12,16)(10,14,13,15)(17,21,22,19,23,33,24,32)(18,27,28,39,29,25,30,42)(20,35,36,48,37,40,38,31)(26,41,34,43,44,46,45,47);(1,4,5,3)(2,7,8,6)(17,22,23,24)(18,28,29,30)(19,21,32,33)(20,36,37,38)(25,39,27,42)(26,34,44,45)(31,40,48,35)(41,47,46,43)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=h,d^2=g,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,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7504411);