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×#550 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1621 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)(19,24)(20,31)(21,36)(22,38)(23,28)(25,35)(26,30)(27,33)(29,41)(32,39)(34,47)(37,40)(42,48)(43,44)(45,46);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,30)(18,24,28,26)(20,32,34,48)(21,22,37,33)(25,41,43,46)(27,36,38,40)(29,44,45,35)(31,39,47,42);(1,3,5,4)(2,7,6,8)(10,13)(14,15)(17,20,33,44,23,34,22,35)(18,25,38,47,28,43,27,31)(19,29,37,48,30,45,21,32)(24,39,36,46,26,42,40,41);(9,11,12,16)(10,14,13,15)(17,21,22,30,23,37,33,19)(18,26,27,36,28,24,38,40)(20,32,35,45,34,48,44,29)(25,42,31,46,43,39,47,41);(1,4,5,3)(2,7,6,8)(20,34)(25,43)(29,45)(31,47)(32,48)(35,44)(39,42)(41,46)Polycyclic presentation:
2^9|b^2=i,c^2=fhi,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,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7504653);