Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6132643 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(11,15)(14,16)(17,18)(19,22)(20,25)(21,24)(23,28)(26,29)(27,31)(30,32)(33,34,39,36)(35,41,37,40)(38,42,46,44)(43,48,45,47);(3,8)(5,7)(10,13)(14,16)(18,19)(21,29)(23,25)(27,32)(33,35,39,37)(34,40,36,41)(38,43,46,45)(42,47,44,48);(9,10,12,13)(11,14,15,16)(17,19,22,18)(20,25,28,23)(21,24,29,26)(27,30,32,31)(33,36,39,34)(35,40,37,41)(38,42,46,44)(43,48,45,47);(1,3)(2,5)(4,8)(6,7)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,37,39,35)(34,41,36,40)(38,45,46,43)(42,48,44,47);(1,2)(3,5)(4,6)(7,8)(9,11,12,15)(10,14,13,16)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,38,39,46)(34,42,36,44)(35,43,37,45)(40,47,41,48)Polycyclic presentation:
2^9|a^2=gi,b^2=fg,c^2=h,d^2=f,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6087484);