Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667836 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2433 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1683 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,5)(4,6)(7,8)(17,18,22,25)(19,26,28,23)(20,29,21,24)(27,31,30,32);(1,3,4,7)(2,5,6,8)(10,11)(12,15)(14,16)(17,19,22,28)(18,23,25,26)(20,27,21,30)(24,31,29,32);(2,6)(5,8)(9,10)(11,15)(12,16)(13,14);(10,14)(11,16)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,26,32);(9,11,13,16)(10,15,14,12)(17,21,22,20)(18,24,25,29)(19,27,28,30)(23,32,26,31)Polycyclic presentation:
2^9|a^2=f,b^2=fg,d^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,d^c=di,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6589345);