Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1452 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(6,8)(11,12)(15,16)(17,18,22,19)(20,25,27,23)(21,24,29,26)(28,31,32,30)(33,34,37,40)(35,41,43,38)(36,39,44,46)(42,47,48,45);(2,5)(6,8)(10,14)(15,16)(18,19)(20,27)(24,26)(28,32)(36,44)(39,46)(42,48)(45,47);(1,2,4,5)(3,6,7,8)(9,10)(11,15)(12,16)(13,14)(17,19,22,18)(20,23,27,25)(21,24,29,26)(28,31,32,30)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,46)(45,48);(9,11,13,12)(10,15,14,16)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31)(33,35,37,43)(34,38,40,41)(36,42,44,48)(39,45,46,47);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,14,15)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,36,37,44)(34,39,40,46)(35,42,43,48)(38,45,41,47)Polycyclic presentation:
2^9|a^2=fg,c^2=fghi,d^2=i,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5238288);