Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394048 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1048 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,7)(3,8,4,6)(11,15)(13,16)(17,18,19,23,22,26,21,25)(20,29,28,24,30,31,27,32);(1,3,5,4)(2,6,7,8)(9,10,12,14)(11,16,15,13)(17,19,22,21)(18,23,26,25)(20,28,30,27)(24,31,32,29);(9,11,12,15)(10,13,14,16)(17,20)(18,24)(19,27)(21,28)(22,30)(23,29)(25,31)(26,32);(1,4,5,3)(2,6,7,8)(17,20)(18,24)(19,28)(21,27)(22,30)(23,31)(25,29)(26,32)Polycyclic presentation:
2^9|a^2=efg,b^2=eg,c^2=gi,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,f^c=fi,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 311097);