Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667900 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1462 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)(3,7)(18,22)(20,27)(23,29)(24,28)(26,31)(30,32);(1,2,4,5)(3,6,7,8)(9,10,12,13)(11,15,16,14);(2,5)(6,8)(10,13)(14,15)(17,18,21,24)(19,23,25,30)(20,28,27,22)(26,32,31,29);(1,3,4,7)(2,6,5,8)(17,19)(18,23)(20,26)(21,25)(22,29)(24,30)(27,31)(28,32);(9,11,12,16)(10,14,13,15)(19,25)(23,30)(26,31)(29,32)Polycyclic presentation:
2^9|b^2=h,c^2=i,d^2=f,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6707984);