Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #234973 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35050 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1897 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10)(3,12,15,24)(4,16,19,8)(6,11,20,9)(7,17,23,14)(13,27,30,28)(18,26,32,25)(21,29,22,31);(1,3)(2,7)(4,17)(5,15)(6,13)(8,24)(9,21)(10,23)(11,22)(12,16)(14,19)(18,31)(20,30)(25,28)(26,27)(29,32);(4,18)(7,21)(8,25)(12,24)(14,29)(16,26)(17,31)(19,32)(22,23)(27,28);(3,13)(7,22)(12,27)(14,29)(15,30)(17,31)(21,23)(24,28)Polycyclic presentation:
2^9|a^2=h,e^2=h,b^a=be,c^a=cf,e^a=eh,g^a=ghi,c^b=cg,d^b=di,e^b=eh,f^b=fh,e^c=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 142170);