Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #198516 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1278 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7,9,16)(3,11,13,30)(4,15)(10,22,21,31)(12,29,17,14)(19,28)(20,27)(24,25);(1,2,4,16,18,21,28,22)(3,8,17,24,29,32,11,27)(5,10,15,31,6,9,19,7)(12,20,14,26,30,25,13,23);(1,3)(2,8)(4,17)(5,14)(6,13)(7,20)(9,23)(10,26)(11,28)(12,19)(15,30)(16,24)(18,29)(21,32)(22,27)(25,31);(2,9)(7,16)(8,23)(10,21)(20,24)(22,31)(25,27)(26,32)Polycyclic presentation:
2^9|a^2=g,b^2=eg,e^2=hi,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=di,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fhi,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 105712);