Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4379698 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(6,8)(11,15)(13,16)(17,18)(19,25)(20,22)(21,28)(23,26)(24,30)(27,31)(29,32)(33,34,37,40)(35,38,41,45)(36,43,44,39)(42,48,47,46);(9,10,12,14)(11,13,15,16)(18,20)(19,26)(24,28)(27,29)(35,41)(38,45)(42,47)(46,48);(1,2,4,5)(3,6,7,8)(17,19)(18,23)(20,25)(21,29)(22,26)(24,32)(27,30)(28,31)(33,35)(34,38)(36,42)(37,41)(39,46)(40,45)(43,48)(44,47);(1,3,4,7)(2,6,5,8)(9,11)(10,13)(12,15)(14,16)(17,20,22,18)(19,25,26,23)(21,28,30,24)(27,32,29,31)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,32)(26,29)(33,36,37,44)(34,39,40,43)(35,42,41,47)(38,46,45,48)Polycyclic presentation:
2^9|a^2=fh,b^2=hi,c^2=fg,d^2=g,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4293227);