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×#2407 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3358 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)(10,13)(11,15)(17,18,22,20)(19,23,25,26)(21,24,30,28)(27,31,29,32);(5,6,7,8)(10,13)(14,16)(18,20)(19,25)(24,28)(27,29);(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,26)(21,29)(22,25)(24,32)(27,30)(28,31);(1,2)(3,4)(9,10)(11,14)(12,13)(15,16)(17,20,22,18)(19,26,25,23)(21,24,30,28)(27,31,29,32);(9,11)(10,14)(12,15)(13,16)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)Polycyclic presentation:
2^9|a^2=f,b^2=fgh,d^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, 3242036);