Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #124 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2398 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
994 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)(7,19,22,36)(8,14,25,34)(9,15)(10,16)(11,18)(12,17)(13,21,31,27)(20,35)(23,29)(24,33,28,30)(26,32);(1,2)(3,4)(5,7,11,22)(6,13)(8,23,25,9)(10,21,12,27)(14,32)(15,30)(16,19)(17,36)(18,31)(20,28,26,24)(29,33)(34,35);(2,4)(5,8)(6,14)(7,20)(9,21)(10,24)(11,25)(12,28)(13,29)(15,31)(16,33)(17,30)(18,34)(19,35)(22,26)(23,27)(32,36);(5,9,10,26,11,23,12,20)(6,15,16,32,18,29,17,35)(7,8,21,24,22,25,27,28)(13,30,19,14,31,33,36,34)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,g^b=gi,h^b=hi,g^c=gi,h^c=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265616);