Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C22×D4 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1315 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)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,25,21,26,20,29)(19,24,27,31,28,23,30,32);(1,3,6,2)(4,7,5,8)(9,11)(10,13)(12,15)(14,16)(17,19,21,28)(18,23,26,24)(20,30,22,27)(25,32,29,31);(10,14)(11,15)(17,19)(18,24)(20,30)(21,28)(22,27)(23,26)(25,31)(29,32);(1,4,6,5)(2,7,3,8)(10,14)(13,16)(17,18,20,29,21,26,22,25)(19,23,30,31,28,24,27,32);(1,5,6,4)(2,7,3,8)(17,20,21,22)(18,25,26,29)(19,27,28,30)(23,31,24,32)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=fh,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cgi,d^b=dfi,e^b=ei,d^c=di,e^c=ei,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6328875);