Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518964 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×D8 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1472 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,5)(3,7,8,6)(9,10,12,14)(11,15,16,13)(17,18)(19,24)(20,32)(21,37)(22,27)(23,28)(25,35)(26,31)(29,42)(30,41)(33,40)(34,43)(36,47)(38,46)(39,45)(44,48);(2,5)(6,7)(11,16)(13,15)(17,19,22,30)(18,24,27,41)(20,33,35,29)(21,38,31,23)(25,42,32,40)(26,28,37,46)(34,48,39,36)(43,44,45,47);(9,11,12,16)(10,13,14,15)(17,20,22,35)(18,25,27,32)(19,29,30,33)(21,39,31,34)(23,36,38,48)(24,40,41,42)(26,45,37,43)(28,44,46,47);(10,14)(13,15)(17,21,23,19,22,31,38,30)(18,26,28,41,27,37,46,24)(20,34,36,33,35,39,48,29)(25,43,44,40,32,45,47,42);(1,3,4,8)(2,6,5,7)(20,35)(25,32)(29,33)(34,39)(36,48)(40,42)(43,45)(44,47)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=g,d^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7522473);