Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518992 |
| 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 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#125 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1408 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)(10,14)(11,12)(17,18,23,19)(20,29,32,25)(21,24,31,28)(22,26,30,27);(2,6)(7,8)(9,10)(11,16)(12,15)(13,14)(17,19,23,18)(20,25,32,29)(21,28,31,24)(22,27,30,26);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,15,14,16)(17,20,30,31,23,32,22,21)(18,24,26,25,19,28,27,29);(9,11,13,12)(10,16,14,15)(17,21,22,32,23,31,30,20)(18,25,27,24,19,29,26,28);(1,4,5,3)(2,7,6,8)(9,12,13,11)(10,15,14,16)(17,22,23,30)(18,26,19,27)(20,31,32,21)(24,29,28,25)Polycyclic presentation:
2^9|a^2=fgh,b^2=i,c^2=fhi,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=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7512442);