Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476659 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2826 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
817 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,12,16,10,18,11,17)(7,19,21,30,23,13,22,26)(8,24,29,34,27,20,28,14)(9,15,33,25,31,35,32,36);(1,2)(3,4)(5,7,12,22,10,23,11,21)(6,13,17,26,18,19,16,30)(8,25,29,15,27,36,28,35)(9,14,33,20,31,34,32,24);(2,4)(5,8,11,28,10,27,12,29)(6,14,16,24,18,34,17,20)(7,15,21,25,23,35,22,36)(9,30,32,19,31,26,33,13);(5,9,10,31)(6,15)(7,20)(8,26,27,30)(11,32,12,33)(13,29,19,28)(14,21)(16,25)(17,36)(18,35)(22,34)(23,24)Polycyclic presentation:
2^9|a^2=ef,b^2=fi,c^2=f,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=def,e^b=ei,g^b=gi,d^c=de,g^c=gi,h^c=hi,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, 385121);