Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 | C2×#127 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1340 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(7,8)(10,13)(11,15)(17,18,22,26,23,27,21,25)(19,28,29,20,30,31,32,24);(1,2,5,6)(3,7,4,8)(10,13)(11,15);(1,3,5,4)(2,7,6,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,27)(21,29)(22,32)(23,30)(25,28)(26,31);(1,4,5,3)(2,7,6,8)(10,13)(14,16)(17,20,23,24)(18,19,27,30)(21,28,22,31)(25,32,26,29);(9,11,12,15)(10,14,13,16)(17,21,23,22)(18,25,27,26)(19,29,30,32)(20,31,24,28)Polycyclic presentation:
2^9|a^2=gh,b^2=f,c^2=f,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7079474);