Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476297 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1016 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,16,11,17,29,15)(7,18,9,12,20,31,27,14)(8,21,24,34,25,33,28,13)(19,35,22,32,23,30,26,36);(1,2)(3,4)(5,7,10,9,11,20,29,27)(6,12,15,18,17,14,16,31)(8,22,24,23,25,26,28,19)(13,32,33,35,34,36,21,30);(2,4)(5,8)(6,13)(7,19)(9,26)(10,28)(11,25)(12,30)(14,35)(15,21)(16,33)(17,34)(18,36)(20,23)(22,27)(24,29)(31,32);(5,9,10,20,11,27,29,7)(6,14,15,31,17,12,16,18)(8,23,24,26,25,19,28,22)(13,32,33,35,34,36,21,30)Polycyclic presentation:
2^9|a^2=f,b^2=e,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=def,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383895);