Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | C2×#2430 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3365 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)(9,10,12,14)(11,15,16,13)(17,18,22,19)(20,25,27,23)(21,24,30,26)(28,32,29,31);(5,6,7,8)(11,16)(13,15)(18,19)(20,27)(24,26)(28,29);(5,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,18)(20,25,27,23)(21,24,30,26)(28,31,29,32);(1,2)(3,4)(9,11)(10,13)(12,16)(14,15)(17,20)(18,23)(19,25)(21,29)(22,27)(24,32)(26,31)(28,30);(9,10)(11,13)(12,14)(15,16)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,32)(27,29)Polycyclic presentation:
2^9|a^2=fg,b^2=fgh,c^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,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, 3242064);