Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| 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 | |
1496 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(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)(35,41)(36,39)(37,40)(38,43)(42,47)(44,45)(46,48);(3,8)(5,7)(10,13)(14,16)(18,19)(21,29)(23,25)(27,32)(33,34,37,40)(35,41,43,38)(36,44,45,39)(42,46,48,47);(9,10)(11,14)(12,13)(15,16)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,45)(46,48);(1,3)(2,5)(4,8)(6,7)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32)(33,35,37,43)(34,38,40,41)(36,42,45,48)(39,46,44,47);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)(33,36,37,45)(34,39,40,44)(35,42,43,48)(38,46,41,47)Polycyclic presentation:
2^9|a^2=fi,b^2=fg,c^2=g,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5895664);