Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4928752 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1881 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(6,8)(9,10)(11,16)(12,13)(14,15)(17,18)(19,22)(20,23)(21,24)(25,26)(27,32)(28,30)(29,31);(10,15)(11,12)(17,19,21,20)(18,22,24,23)(26,28)(31,32);(1,2,4,5)(3,6,7,8)(9,11,14,12)(10,16,15,13)(17,18,21,24)(19,22,20,23)(25,27,30,29)(26,31,28,32);(1,2)(3,6)(4,5)(7,8)(9,12,14,11)(10,13,15,16)(17,18)(19,22)(20,23)(21,24)(25,28,30,26)(27,31,29,32);(1,3,4,7)(2,6,5,8)(9,13)(10,11)(12,15)(14,16)(17,20,21,19)(18,23,24,22)(25,29,30,27)(26,31,28,32)Polycyclic presentation:
2^9|b^2=gi,c^2=gh,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,d^c=dg104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4836822);