Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(6,8)(11,15)(14,16)(17,18)(19,25)(20,24)(21,22)(23,26)(27,32)(28,31)(29,30)(33,34,37,40)(35,41,43,38)(36,44,45,39)(42,46,48,47);(10,13)(14,16)(18,22)(19,26)(24,30)(27,28)(33,34,37,40)(35,41,43,38)(36,39,45,44)(42,47,48,46);(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,28)(21,26)(22,25)(24,32)(27,29)(30,31)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,45)(47,48);(1,2)(3,6)(4,5)(7,8)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,32)(29,30)(33,35,37,43)(34,38,40,41)(36,42,45,48)(39,46,44,47);(1,3,4,7)(2,6,5,8)(9,11,12,15)(10,14,13,16)(17,20,21,29)(18,24,22,30)(19,27,26,28)(23,31,25,32)(33,36,37,45)(34,39,40,44)(35,42,43,48)(38,46,41,47)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,d^2=fg,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4654679);