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×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1399 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(9,10,13,12)(11,14,16,15)(17,18,24,29)(19,28,32,40)(20,26,35,44)(21,27,23,38)(22,37,31,25)(30,42,46,48)(33,41,36,43)(34,47,45,39);(1,3,4,8)(2,5,6,7)(9,11)(10,14)(12,15)(13,16)(17,19,23,31,24,32,21,22)(18,25,27,28,29,37,38,40)(20,30,33,45,35,46,36,34)(26,39,43,42,44,47,41,48);(2,6)(3,8)(9,12,13,10)(11,15,16,14)(17,20,24,35)(18,26,29,44)(19,30,32,46)(21,36,23,33)(22,34,31,45)(25,39,37,47)(27,43,38,41)(28,42,40,48);(2,6)(3,8)(11,16)(14,15)(17,21,24,23)(18,27,29,38)(19,31,32,22)(20,33,35,36)(25,40,37,28)(26,41,44,43)(30,34,46,45)(39,42,47,48);(2,6)(5,7)(17,22,21,32,24,31,23,19)(18,28,38,25,29,40,27,37)(20,34,36,46,35,45,33,30)(26,42,41,39,44,48,43,47)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=gh,d^2=i,e^2=fgi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,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, 7514588);