Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453996 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
871 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,6)(3,5,7,8)(10,13)(14,15)(18,23)(19,27)(20,21)(24,31)(25,26)(28,29);(1,3)(2,5)(4,7)(6,8)(9,10,12,13)(11,15,16,14);(3,7)(5,8)(9,11)(10,14)(12,16)(13,15)(17,18,20,23,22,26,21,25)(19,24,27,30,28,31,29,32);(11,16)(14,15)(17,19)(18,24)(20,27)(21,29)(22,28)(23,30)(25,32)(26,31)Polycyclic presentation:
2^9|a^2=eg,b^2=e,c^2=f,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^c=de,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 368682);