Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996335 |
| 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 | C22×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#125 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1447 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)(6,7)(10,14)(12,15)(17,18,21,24)(19,25,27,32)(20,29,26,23)(22,30,31,28);(1,2,4,5)(3,7,8,6)(9,10,13,14)(11,12,16,15);(9,11)(10,15)(12,14)(13,16)(17,19,26,30,21,27,20,28)(18,22,29,32,24,31,23,25);(1,3,4,8)(2,6,5,7)(10,14)(11,16)(19,27)(22,31)(25,32)(28,30);(9,12,13,15)(10,11,14,16)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,d^2=fh,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7106413);