Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5955303 |
| automorphism | Aut(G) order 8388608 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1914 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(10,13)(11,15)(18,20)(19,23)(25,26,29,32)(27,30,31,28);(2,5)(6,8)(9,10,12,13)(11,14,15,16)(19,23)(22,24)(27,31)(28,30);(9,10)(11,14)(12,13)(15,16)(17,18,21,20)(19,22,23,24)(25,27,29,31)(26,30,32,28);(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,22)(20,24)(21,23)(25,28,29,30)(26,31,32,27);(1,3)(2,6)(4,7)(5,8)(9,11,12,15)(10,14,13,16)(17,20,21,18)(19,24,23,22)(25,26)(27,30)(28,31)(29,32)Polycyclic presentation:
2^9|a^2=fh,b^2=f,c^2=gh,d^2=gi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5863580);