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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1406 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(3,4)(9,10)(11,15)(12,16)(13,14)(17,18)(19,29)(20,25)(21,27)(22,26)(23,28)(24,31)(30,45)(32,39)(33,42)(34,40)(35,43)(36,41)(37,44)(38,46)(47,48);(1,2)(3,7)(4,8)(5,6)(17,19,22,31)(18,24,26,29)(20,33,32,23)(21,34,36,35)(25,28,39,42)(27,43,41,40)(30,37,46,47)(38,44,45,48);(1,3,5,4)(2,7,6,8)(10,14)(15,16)(17,20,23,19,22,32,33,31)(18,25,28,29,26,39,42,24)(21,35,37,46,36,34,47,30)(27,43,44,38,41,40,48,45);(1,4,5,3)(2,8,6,7)(9,11,13,12)(10,15,14,16)(18,26)(21,36)(24,29)(25,39)(28,42)(30,46)(34,35)(37,47);(2,6)(7,8)(9,12,13,11)(10,16,14,15)(17,21,33,47,22,36,23,37)(18,27,42,48,26,41,28,44)(19,30,20,34,31,46,32,35)(24,38,39,43,29,45,25,40)Polycyclic presentation:
2^9|b^2=i,c^2=fhi,d^2=g,e^2=fgh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6751117);