Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #34889 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1685 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(3,4)(6,8)(9,10,12,13)(11,15,16,14)(17,18,20,23,21,24,27,30)(19,22,26,29,25,28,31,32);(1,2)(3,8)(4,6)(5,7)(9,10)(11,14)(12,13)(15,16)(19,25)(22,28)(26,31)(29,32);(1,3,5,4)(2,6,7,8)(10,13)(14,15);(1,4,5,3)(2,6,7,8)(10,13)(14,15)(17,19,20,26,21,25,27,31)(18,22,23,29,24,28,30,32);(9,11,12,16)(10,14,13,15)Polycyclic presentation:
2^9|a^2=h,c^2=fg,d^2=gh,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dfi128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6332977);