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 | #42664 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1459 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)(3,8)(4,6)(5,7)(10,14)(11,12)(17,18)(19,24)(20,32)(21,37)(22,27)(23,28)(25,35)(26,31)(29,45)(30,40)(33,43)(34,42)(36,48)(38,41)(39,47)(44,46);(9,10)(11,16)(12,15)(13,14)(17,19,22,30)(18,24,27,40)(20,33,35,38)(21,34,31,23)(25,41,32,43)(26,28,37,42)(29,46,47,36)(39,48,45,44);(1,3,5,4)(2,6,7,8)(9,11,13,12)(10,15,14,16)(17,20,34,46,22,35,23,36)(18,25,42,48,27,32,28,44)(19,29,21,38,30,47,31,33)(24,39,37,43,40,45,26,41);(9,12,13,11)(10,15,14,16)(20,35)(25,32)(29,47)(33,38)(36,46)(39,45)(41,43)(44,48);(1,4,5,3)(2,6,7,8)(17,21,23,19,22,31,34,30)(18,26,28,40,27,37,42,24)(20,33,36,47,35,38,46,29)(25,43,44,39,32,41,48,45)Polycyclic presentation:
2^9|b^2=i,c^2=fgh,d^2=f,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,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, 6744710);