Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438991 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1576 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,8,4,6)(9,10)(11,13)(12,14)(15,16)(17,18,23,25)(19,22,27,21)(20,24,29,32)(26,28,31,30);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16);(10,14)(11,15)(17,19,22,25,23,27,21,18)(20,26,28,32,29,31,30,24);(1,4,5,3)(2,6,7,8)(11,15)(13,16)(17,20)(18,24)(19,26)(21,30)(22,28)(23,29)(25,32)(27,31);(17,21,23,22)(18,19,25,27)(20,28,29,30)(24,31,32,26)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=gh,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^c=ei,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6367463);