Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #238414 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#393 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
988 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,6)(4,11,13,7)(5,8)(9,14,12,16)(10,15)(17,18)(19,22)(20,23)(21,25)(24,29)(26,31)(27,30)(28,32);(1,3)(2,6)(4,12,13,9)(5,10)(7,16,11,14)(8,15)(17,19)(18,22)(20,28)(21,26)(23,32)(24,30)(25,31)(27,29);(1,4)(2,7)(3,9)(5,13)(6,14)(8,11)(10,12)(15,16)(18,23)(20,29)(22,30)(24,25)(27,28)(31,32);(19,26)(22,31)(27,28)(30,32)Polycyclic presentation:
2^9|a^2=e,b^2=e,f^2=hi,b^a=be,c^a=cf,f^a=fi,g^a=ghi,c^b=cg,d^b=dhi,f^b=fi,g^b=gh,e^c=eh,f^c=fhi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 145610);