Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #845297 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C26 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2324 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(3,7)(6,8)(10,14)(15,16)(18,20)(22,23)(25,26)(27,28)(29,31)(30,32);(1,2,5,4)(3,6,7,8)(9,10)(11,15)(12,16)(13,14)(17,18,21,20)(19,23,24,22)(25,27,29,32)(26,28,31,30);(9,11,13,12)(10,15,14,16)(17,18,21,20)(19,22,24,23)(25,26,29,31)(27,30,32,28);(1,3)(2,6)(4,8)(5,7)(9,12,13,11)(10,16,14,15)(17,19,21,24)(18,22,20,23)(25,27)(26,30)(28,31)(29,32);(1,4,5,2)(3,8,7,6)(9,10)(11,15)(12,16)(13,14)(17,20,21,18)(19,23,24,22)(25,28,29,30)(26,27,31,32)Polycyclic presentation:
2^9|b^2=ghi,c^2=fh,d^2=f,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 807185);