Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | |
1466 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)(3,5,7,8)(9,10,13,12)(11,14,15,16)(17,18,22,19)(20,25,28,23)(21,24,30,26)(27,32,29,31)(34,35)(36,43)(37,45)(47,48);(11,15)(14,16)(18,19)(23,25)(24,26)(31,32)(33,34,38,35)(36,39,43,41)(37,42,45,40)(44,48,46,47);(9,11)(10,14)(12,16)(13,15)(17,19,22,18)(20,25,28,23)(21,24,30,26)(27,31,29,32)(33,35,38,34)(36,39,43,41)(37,40,45,42)(44,48,46,47);(1,2)(3,5)(4,6)(7,8)(17,20)(18,23)(19,25)(21,29)(22,28)(24,32)(26,31)(27,30)(33,36)(34,39)(35,41)(37,46)(38,43)(40,48)(42,47)(44,45);(1,3,4,7)(2,5,6,8)(9,12,13,10)(11,16,15,14)(17,21)(18,24)(19,26)(20,27)(22,30)(23,31)(25,32)(28,29)(33,37,38,45)(34,40,35,42)(36,44,43,46)(39,47,41,48)Polycyclic presentation:
2^9|a^2=fhi,b^2=g,c^2=f,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4185658);