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 | C2×#2688 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2906 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)(11,15)(12,13)(14,16)(17,18,22,25)(19,20,27,23)(21,24,31,32)(26,28,29,30);(6,8)(11,16)(14,15)(17,19,22,27)(18,23,25,20)(21,26,31,29)(24,30,32,28);(5,6)(7,8)(10,13)(14,15)(17,19)(18,20)(21,29)(22,27)(23,25)(24,30)(26,31)(28,32);(1,2)(3,4)(9,10)(11,14)(12,13)(15,16)(17,20)(18,19)(21,30)(22,23)(24,29)(25,27)(26,32)(28,31);(9,11)(10,14)(12,16)(13,15)(17,21)(18,24)(19,26)(20,28)(22,31)(23,30)(25,32)(27,29)Polycyclic presentation:
2^9|a^2=f,b^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, 4038428);