Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1422 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,7,4,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,27)(20,28)(21,24)(22,26)(23,30)(25,31)(29,32);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,21)(18,24,26,27)(20,25,30,32)(23,31,28,29);(1,3,5,4)(2,7,6,8)(10,13)(14,15)(17,20,28,18,22,30,23,26)(19,27,31,32,21,24,29,25);(9,11,12,16)(10,14,13,15)(17,21,23,31,22,19,28,29)(18,25,20,24,26,32,30,27);(1,4,5,3)(2,7,6,8)(18,26)(20,30)(24,27)(25,32)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=fh,d^2=g,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7504954);