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 | #2388 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1433 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,29)(19,25,32,40)(20,33,36,26)(21,27,24,30)(22,28,38,44)(31,45,46,39)(34,47,48,41)(35,43,37,42);(1,3)(2,5)(4,8)(6,7)(9,11,12,15)(10,13,14,16)(17,19,23,32)(18,25,29,40)(20,34,36,48)(21,22,24,38)(26,41,33,47)(27,28,30,44)(31,35,46,37)(39,42,45,43);(10,14)(13,16)(17,20,24,37,23,36,21,35)(18,26,30,43,29,33,27,42)(19,31,22,34,32,46,38,48)(25,39,28,41,40,45,44,47);(2,6)(5,7)(17,21,23,24)(18,27,29,30)(19,22,32,38)(20,35,36,37)(25,28,40,44)(26,42,33,43)(31,34,46,48)(39,41,45,47);(3,8)(5,7)(17,22,24,32,23,38,21,19)(18,28,30,40,29,44,27,25)(20,34,37,46,36,48,35,31)(26,41,43,45,33,47,42,39)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=h,d^2=i,e^2=h,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, 6719421);