Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 | #42880 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1622 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)(17,18,23,29)(19,30,32,25)(20,26,35,44)(21,37,24,27)(22,28,38,40)(31,45,46,39)(33,43,47,41)(34,48,36,42);(1,2,4,5)(3,6,7,8)(17,19)(18,25)(20,33)(21,22)(23,32)(24,38)(26,41)(27,28)(29,30)(31,36)(34,46)(35,47)(37,40)(39,42)(43,44)(45,48);(9,11,13,12)(10,14,15,16)(17,20,24,36,23,35,21,34)(18,26,27,42,29,44,37,48)(19,31,22,33,32,46,38,47)(25,39,40,41,30,45,28,43);(1,3,4,7)(2,6,5,8)(17,21,23,24)(18,27,29,37)(19,22,32,38)(20,34,35,36)(25,28,30,40)(26,42,44,48)(31,33,46,47)(39,43,45,41);(2,5)(6,8)(9,12,13,11)(10,14,15,16)(17,22,24,32,23,38,21,19)(18,28,27,25,29,40,37,30)(20,33,36,46,35,47,34,31)(26,43,42,39,44,41,48,45)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=gh,d^2=f,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6744169);