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 | |
1432 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)(19,27)(20,34)(21,40)(22,30)(23,29)(24,28)(25,33)(26,39)(31,43)(32,42)(35,41)(36,46)(37,45)(38,44)(47,48);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,33)(18,25,29,27)(20,35,38,36)(21,22,32,24)(26,43,45,48)(28,42,30,40)(31,39,47,37)(34,46,44,41);(10,13)(14,15)(17,20,24,39,23,38,22,37)(18,26,30,34,29,45,28,44)(19,31,32,36,33,47,21,35)(25,41,42,48,27,46,40,43);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,21,22,33,23,32,24,19)(18,27,28,42,29,25,30,40)(20,36,37,31,38,35,39,47)(26,41,44,43,45,46,34,48);(1,4,5,3)(2,7,6,8)(17,22,23,24)(18,28,29,30)(19,32,33,21)(20,37,38,39)(25,42,27,40)(26,44,45,34)(31,36,47,35)(41,48,46,43)Polycyclic presentation:
2^9|a^2=fi,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=cgi,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, 7505496);