Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573485 |
| 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 | #1999 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#126 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1437 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,24)(20,26)(21,30)(22,25)(23,27)(28,32)(29,31);(1,3,5,4)(2,6,7,8)(17,19,22,20)(18,24,25,26)(21,29,23,28)(27,32,30,31);(10,13)(14,16)(18,25)(19,28)(20,29)(21,23)(24,31)(26,32);(9,11,12,15)(10,14,13,16)(17,20,22,19)(18,26,25,24)(21,28,23,29)(27,31,30,32);(1,4,5,3)(2,6,7,8)(17,21)(18,27)(22,23)(24,26)(25,30)(28,29)Polycyclic presentation:
2^9|b^2=fi,d^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6494764);