Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476293 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
897 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,14,11,16,10,15)(7,17,20,12,21,29,19,28)(8,22,25,31,26,13,24,32)(18,30,23,27,34,33,35,36);(1,2)(3,4)(5,7,9,20,11,21,10,19)(6,12,15,17,16,28,14,29)(8,23,25,34,26,35,24,18)(13,30,31,36,22,33,32,27);(2,4)(5,8)(6,13)(7,18)(9,24)(10,25)(11,26)(12,27)(14,31)(15,32)(16,22)(17,33)(19,23)(20,35)(21,34)(28,36)(29,30);(6,14)(7,19)(8,24)(9,10)(15,16)(17,29)(20,21)(23,35)(25,26)(27,33)(30,36)(31,32)Polycyclic presentation:
2^9|a^2=ef,b^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=def,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,g^d=gi,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383631);