Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4311730 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3160 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(6,7)(11,15)(14,16)(17,18,22,25)(19,26,23,20)(21,29,31,24)(27,28,30,32);(10,13)(14,16)(17,19,22,23)(18,20,25,26)(21,30,31,27)(24,32,29,28);(5,6,8,7)(9,10,12,13)(11,14,15,16)(17,19)(18,20)(21,27)(22,23)(24,28)(25,26)(29,32)(30,31);(1,2,3,4)(5,7,8,6)(17,20)(18,23)(19,25)(21,28)(22,26)(24,30)(27,29)(31,32);(9,11)(10,14)(12,15)(13,16)(17,21)(18,24)(19,27)(20,28)(22,31)(23,30)(25,29)(26,32)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=gh,d^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4248995);