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 | |
1453 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)(3,7)(9,10)(11,14)(12,16)(13,15)(17,18,22,20)(19,25,27,23)(21,28,30,24)(26,31,29,32)(33,34)(35,37)(36,43)(38,39)(40,45)(41,46)(42,48)(44,47);(3,7)(6,8)(9,11,13,12)(10,14,15,16)(18,20)(23,25)(24,28)(31,32)(34,39)(35,41)(40,43)(42,47);(1,2,4,5)(3,6,7,8)(9,12,13,11)(10,14,15,16)(17,19,22,27)(18,23,20,25)(21,29,30,26)(24,32,28,31)(33,35,38,41)(34,37,39,46)(36,42,45,47)(40,44,43,48);(1,3)(2,6)(4,7)(5,8)(9,10)(11,16)(12,14)(13,15)(17,20,22,18)(19,25,27,23)(21,28,30,24)(26,32,29,31)(33,36,38,45)(34,40,39,43)(35,42,41,47)(37,44,46,48);(17,21,22,30)(18,24,20,28)(19,26,27,29)(23,31,25,32)(33,37)(34,35)(36,44)(38,46)(39,41)(40,42)(43,47)(45,48)Polycyclic presentation:
2^9|a^2=fh,b^2=fgi,c^2=gh,d^2=f,e^2=fh,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, 4533559);