Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476613 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2826 |
| 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 | |
829 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,9,15,11,16,10,14)(7,17,20,12,19,27,21,28)(8,22,24,31,25,30,23,13)(18,29,33,32,35,26,34,36);(5,7,9,20,11,19,10,21)(6,12,14,17,16,28,15,27)(8,18,24,33,25,35,23,34)(13,29,30,36,31,26,22,32);(5,8,9,24,11,25,10,23)(6,13,14,30,16,31,15,22)(7,18,20,33,19,35,21,34)(12,26,17,32,28,29,27,36);(2,4)(6,14)(7,19)(8,23)(9,10)(12,27)(15,16)(17,28)(18,33)(22,30)(24,25)(26,29)(34,35)Polycyclic presentation:
2^9|a^2=eg,b^2=e,c^2=e,e^2=i,f^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,f^b=fi,h^b=hi,d^c=de,e^d=ei,f^d=fi,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 385524);