Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419839 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23439 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1580 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)(5,6,10,15,9,16,11,13)(7,12,17,26,19,27,20,31)(8,14,23,29,22,30,24,28)(18,33,32,36,21,25,34,35);(5,7)(6,12)(8,21)(9,19)(10,20)(11,17)(13,26)(14,25)(15,31)(16,27)(18,22)(23,32)(24,34)(28,35)(29,36)(30,33);(2,4)(6,13)(7,17)(10,11)(14,28)(15,16)(18,32)(19,20)(21,34)(23,24)(26,31)(29,30)(35,36);(5,8)(6,14,16,30)(7,18)(9,22)(10,23)(11,24)(12,25,27,33)(13,28,15,29)(17,32)(19,21)(20,34)(26,35,31,36)Polycyclic presentation:
2^9|a^2=f,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327036);