Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24985 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2420 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1268 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)(7,17)(8,22,25,13)(9,14)(10,15)(11,16)(12,19)(18,30,34,33)(20,29)(21,28)(23,36,35,27)(24,32,26,31);(2,4)(5,7,10,20,11,21,9,19)(6,12,14,28,16,29,15,17)(8,23,24,34,25,35,26,18)(13,30,32,27,22,33,31,36);(5,8)(6,13)(7,18)(9,24)(10,26)(11,25)(12,27)(14,31)(15,32)(16,22)(17,33)(19,23)(20,35)(21,34)(28,30)(29,36);(2,4)(6,14,16,15)(12,28,29,17)(13,31,22,32)(27,30,36,33)Polycyclic presentation:
2^9|a^2=h,b^2=fg,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 269680);