Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32941 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C4×#121 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1487 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,15,11,16,9,12)(7,17,20,13,21,28,19,29)(8,22,24,30,25,26,23,14)(18,27,33,35,34,36,32,31);(6,12,16,15)(13,17,29,28)(14,26,30,22)(27,31,36,35);(2,4)(5,7,9,19,11,21,10,20)(6,13,15,28,16,29,12,17)(8,18,23,32,25,34,24,33)(14,27,22,35,30,36,26,31);(5,8)(6,14)(7,18)(9,23)(10,24)(11,25)(12,26)(13,27)(15,22)(16,30)(17,31)(19,32)(20,33)(21,34)(28,35)(29,36)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,g^a=gi,g^d=gi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300064);