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)(17,18,21,22)(19,25,27,23)(20,28,29,24)(26,31,30,32);(5,6,7,8)(9,10,13,15)(11,12,16,14)(18,22)(20,29)(23,25)(26,30);(5,6)(7,8)(9,11,13,16)(10,14,15,12)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,31)(30,32);(1,2)(3,4)(9,11)(10,12)(13,16)(14,15)(17,19,21,27)(18,23,22,25)(20,30,29,26)(24,32,28,31);(9,12)(10,11)(13,14)(15,16)(17,20,21,29)(18,24,22,28)(19,26,27,30)(23,31,25,32)Polycyclic presentation:
2^9|a^2=g,b^2=fh,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, 4040183);