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) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2350 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)(9,10,13,15)(11,12,16,14)(17,18,22,23)(19,26,29,24)(20,21,31,25)(27,32,30,28);(5,6,7,8)(9,10,13,15)(11,12,16,14)(18,23)(21,25)(24,26)(28,32);(5,6)(7,8)(9,11,13,16)(10,14,15,12)(17,19)(18,24)(20,30)(21,32)(22,29)(23,26)(25,28)(27,31);(1,2)(3,4)(9,11)(10,12)(13,16)(14,15)(17,20,22,31)(18,25,23,21)(19,27,29,30)(24,32,26,28);(9,12)(10,11)(13,14)(15,16)(17,21,22,25)(18,20,23,31)(19,28,29,32)(24,27,26,30)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fg,d^2=f,e^2=f,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, 4040181);