Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417581 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1024 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,15)(12,14)(13,16)(17,18,20,22)(19,21,23,24)(26,29)(27,31)(30,32);(1,3,5,8)(2,6,4,7)(9,11,12,16)(10,13,14,15)(17,19)(18,21)(20,23)(22,24);(1,2,5,4)(3,6,8,7)(10,14)(13,15)(19,23)(21,24);(1,4,5,2)(3,6,8,7)(11,16)(13,15)(25,26,28,30)(27,32,31,29)Polycyclic presentation:
2^9|a^2=efg,b^2=e,c^2=ef,d^2=efi,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 344551);