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) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2764 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)(10,12)(11,15)(17,18)(19,26)(20,24)(21,29)(22,23)(25,30)(27,28)(31,32);(5,6,7,8)(10,12)(14,16)(18,23)(19,24)(21,30)(28,32);(5,6)(7,8)(9,10,13,12)(11,16,15,14)(17,19)(18,20)(21,31)(22,24)(23,26)(25,32)(27,30)(28,29);(1,2)(3,4)(9,11,13,15)(10,14,12,16)(17,20)(18,24)(19,23)(21,32)(22,26)(25,27)(28,30)(29,31);(9,12,13,10)(11,16,15,14)(17,21)(18,25)(19,27)(20,28)(22,30)(23,29)(24,31)(26,32)Polycyclic presentation:
2^9|b^2=gh,c^2=fg,d^2=fg,e^2=fg,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, 4040981);