Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1890 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,7,8,6)(9,10,14,12)(11,16,13,15)(17,18)(19,22)(20,21)(23,24)(25,26)(27,29)(28,31)(30,32);(2,4)(6,7)(9,11)(10,15)(12,16)(13,14)(18,21)(22,23)(25,26,28,31)(27,29,32,30);(2,4)(6,7)(11,13)(15,16)(17,18,20,21)(19,22,24,23)(25,27,28,32)(26,29,31,30);(1,3,5,8)(2,6,4,7)(9,12,14,10)(11,15,13,16)(17,19)(18,22)(20,24)(21,23)(25,26)(27,29)(28,31)(30,32);(1,4,5,2)(3,6,8,7)(9,13,14,11)(10,16,12,15)(17,18)(19,23)(20,21)(22,24)(25,27)(26,30)(28,32)(29,31)Polycyclic presentation:
2^9|a^2=gh,b^2=fhi,c^2=fgh,d^2=gh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3814415);