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 | #2598 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
987 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)(7,17,19,28,20,12,24,27)(8,21,23,31,22,30,25,13)(18,32,34,26,33,36,35,29);(1,2)(3,4)(5,7,10,19,11,20,9,24)(6,12,14,27,15,17,16,28)(8,18,23,35,22,33,25,34)(13,29,21,36,31,26,30,32);(2,4)(5,8)(6,13)(7,18)(9,25)(10,23)(11,22)(12,26)(14,21)(15,31)(16,30)(17,29)(19,35)(20,33)(24,34)(27,32)(28,36);(6,14,15,16)(8,22)(12,27,17,28)(13,30,31,21)(18,33)(23,25)(26,36,29,32)(34,35)Polycyclic presentation:
2^9|a^2=f,b^2=eh,d^2=e,f^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,d^c=di,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72321741 |
| maximal normal subgroup of | #72321741 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 313312);