Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454039 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60371 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2161 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,12,18)(7,19,24,31)(8,25,28,34)(9,29,20,15)(10,17,11,16)(13,23,32,22)(14,21,35,27)(26,36,30,33);(1,2)(3,4)(5,7)(6,13)(8,26)(9,21)(10,23)(11,22)(12,24)(14,33)(15,25)(16,31)(17,19)(18,32)(20,27)(28,30)(29,34)(35,36);(2,4)(5,8)(6,14)(7,20)(9,24)(10,27)(11,21)(12,28)(13,15)(16,25)(17,34)(18,35)(19,33)(22,26)(23,30)(29,32)(31,36);(5,9)(6,15)(7,21)(8,22)(10,30)(11,26)(12,20)(13,25)(14,31)(16,33)(17,36)(18,29)(19,35)(23,28)(24,27)(32,34)Polycyclic presentation:
2^9|a^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,g^d=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361236);