Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32700 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2760 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14572 |
| prime radical | #23443 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1382 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12,6,11,18,30)(3,13,16,31,15,8,25,19)(4,17,21,32,20,9,27,28)(5,22,23,14,24,10,26,29);(1,3)(2,8)(4,14)(5,9)(6,15)(7,25)(10,27)(11,13)(12,31)(16,18)(17,24)(19,30)(20,29)(21,22)(23,32)(26,28);(1,4,18,27,6,20,7,21)(2,9,12,28,11,17,30,32)(3,14,25,22,15,29,16,10)(5,19,23,13,24,31,26,8);(1,5)(2,10)(3,9)(4,19)(6,24)(7,26)(8,21)(11,22)(12,14)(13,27)(15,17)(16,32)(18,23)(20,31)(25,28)(29,30)Polycyclic presentation:
2^9|a^2=h,c^2=egh,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,f^b=fi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 297615);