Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| 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 | #34829 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1629 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(3,4)(6,8)(9,10,12,14)(11,15,16,13)(17,18,21,24,20,22,26,29)(19,23,27,30,25,28,31,32);(1,2)(3,8)(4,6)(5,7)(9,11)(10,13)(12,16)(14,15)(18,22)(23,28)(24,29)(30,32);(1,3,5,4)(2,6,7,8)(10,14)(11,16)(18,22)(19,25)(24,29)(27,31);(1,4,5,3)(2,6,7,8)(10,14)(11,16)(17,19,26,31,20,25,21,27)(18,23,29,32,22,28,24,30);(10,14)(13,15)(19,25)(23,28)(27,31)(30,32)Polycyclic presentation:
2^9|a^2=h,c^2=fg,d^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=efi,c^b=cgi,d^b=dfi,d^c=di,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6332831);