Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #478213 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2842 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
834 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)(5,6,11,15,12,17,10,16)(7,18,22,32,23,31,21,13)(8,9,25,29,27,30,26,14)(19,20,24,33,35,28,34,36);(5,7,10,21,12,23,11,22)(6,13,15,18,17,32,16,31)(8,24,25,35,27,34,26,19)(9,28,14,33,30,20,29,36);(5,8,10,25,12,27,11,26)(6,14,15,30,17,29,16,9)(7,19,21,24,23,35,22,34)(13,28,18,33,32,20,31,36);(2,4)(5,9,10,29,12,30,11,14)(6,8,16,25,17,27,15,26)(7,20,22,36,23,28,21,33)(13,24,18,19,32,34,31,35)Polycyclic presentation:
2^9|a^2=efg,b^2=e,c^2=e,d^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=def,f^b=fi,g^b=gi,d^c=de,g^c=gi,h^c=hi,e^d=ei,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386475);