Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1909 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C2×C4 | C43 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | C22⋊#95 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1806 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,10,14,11,15,9,23)(7,16,20,26,19,31,18,12)(8,13,21,28,22,29,24,27)(17,32,33,25,34,30,35,36);(5,7)(6,12,15,26)(8,17)(9,18)(10,20)(11,19)(13,25,29,36)(14,16,23,31)(21,33)(22,34)(24,35)(27,32,28,30);(1,2,3,4)(5,8)(6,13)(7,17)(9,24)(10,21)(11,22)(12,25)(14,28)(15,29)(16,30)(18,35)(19,34)(20,33)(23,27)(26,36)(31,32)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=di,g^a=gi,e^b=ei,d^c=di116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30680);