Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395174 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2482 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1022 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,16,11,15,10,14)(7,17,20,12,19,27,21,28)(8,22,23,31,24,13,25,30)(18,33,35,26,34,32,36,29);(2,4)(5,7,10,21,11,19,9,20)(6,12,16,27,15,28,14,17)(8,18,23,35,24,34,25,36)(13,29,31,32,22,26,30,33);(1,2,3,4)(5,8)(6,13)(7,18)(9,25)(10,23)(11,24)(12,26)(14,30)(15,22)(16,31)(17,32)(19,34)(20,36)(21,35)(27,33)(28,29);(6,14,15,16)(7,19)(12,27,28,17)(13,30,22,31)(18,34)(20,21)(26,33,29,32)(35,36)Polycyclic presentation:
2^9|a^2=ef,b^2=h,c^2=fg,d^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=di,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72252628, #72256865, #72256905 |
| maximal normal subgroup of | #72252628, #72256865, #72256905 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 317890);