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 | #2537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1447 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)(17,18,23,19)(20,27,25,22)(21,29,31,28)(24,32,26,30);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,18)(20,22,25,27)(21,28,31,29)(24,30,26,32);(1,3,5,4)(2,7,6,8)(10,13)(14,15)(17,20,22,18,23,25,27,19)(21,28,30,24,31,29,32,26);(9,11,12,16)(10,14,13,15)(17,21,27,32,23,31,22,30)(18,24,20,28,19,26,25,29);(1,4,5,3)(2,7,6,8)(17,22,23,27)(18,20,19,25)(21,30,31,32)(24,28,26,29)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=fh,d^2=gi,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, 7504956);