Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667652 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #36596 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1702 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(9,10,11,14,12,15,16,13)(17,18)(19,27)(20,24)(21,26)(22,25)(23,29)(28,32)(30,31);(1,2,4,5)(3,7,8,6)(17,19)(18,23)(20,28)(21,30)(22,29)(24,31)(25,27)(26,32);(2,5)(6,7)(10,13)(11,16)(14,15)(17,20,22,21)(18,24,25,26)(19,28,29,30)(23,31,27,32);(18,25)(23,27)(24,26)(31,32);(1,3,4,8)(2,6,5,7)(17,21,22,20)(18,26,25,24)(19,28,29,30)(23,31,27,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=f,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,g^c=gi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6710036);