Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806647 |
| 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 | #42800 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1603 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(9,10)(11,16)(12,14)(13,15)(17,18)(19,29)(20,32)(21,27)(22,26)(23,28)(24,31)(25,35)(30,39)(33,41)(34,42)(36,44)(37,48)(38,40)(43,46)(45,47);(1,2,4,5)(3,6,7,8)(17,19,22,31)(18,24,26,29)(20,33,35,36)(21,23,38,34)(25,41,32,44)(27,42,40,28)(30,46,47,37)(39,48,45,43);(9,11,13,12)(10,14,15,16)(17,20,34,46,22,35,23,37)(18,25,42,48,26,32,28,43)(19,30,38,36,31,47,21,33)(24,39,27,44,29,45,40,41);(1,3,4,7)(2,6,5,8)(18,26)(20,35)(24,29)(27,40)(28,42)(30,47)(33,36)(37,46);(2,5)(6,8)(9,12,13,11)(10,14,15,16)(17,21,34,19,22,38,23,31)(18,27,42,29,26,40,28,24)(20,36,46,47,35,33,37,30)(25,41,48,39,32,44,43,45)Polycyclic presentation:
2^9|b^2=f,c^2=gh,d^2=fi,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6744121);