Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396117 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2420 |
| prime radical | #23438 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1403 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)(5,6,9,16,11,15,10,14)(7,17,19,29,20,12,25,28)(8,21,23,31,24,32,26,13)(18,30,22,27,35,36,34,33);(5,7)(6,12)(8,22)(9,25)(10,19)(11,20)(13,30)(14,28)(15,17)(16,29)(18,23)(21,33)(24,34)(26,35)(27,32)(31,36);(2,4)(5,8)(6,13)(7,18)(9,26)(10,23)(11,24)(12,27)(14,21)(15,31)(16,32)(17,33)(19,34)(20,35)(22,25)(28,30)(29,36);(6,14,15,16)(12,28,17,29)(13,21,31,32)(27,30,33,36)Polycyclic presentation:
2^9|a^2=f,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^b=gi,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303313);