Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24808 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2416 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1044 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6,11,16,12,18,10,17)(7,19,23,34,24,36,22,13)(8,25,27,35,28,21,33,14)(9,29,31,26,32,20,30,15);(1,3,4,2)(5,7,11,23,12,24,10,22)(6,13,17,36,18,34,16,19)(8,26,33,29,28,15,27,20)(9,21,31,14,32,25,30,35);(5,8,10,27,12,28,11,33)(6,14,17,25,18,35,16,21)(7,20,23,26,24,29,22,15)(9,13,30,19,32,34,31,36);(5,9,10,30,12,32,11,31)(6,15,16,29,18,26,17,20)(7,21,22,35,24,25,23,14)(8,13,27,19,28,34,33,36)Polycyclic presentation:
2^9|a^2=h,b^2=efh,c^2=g,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,f^c=fi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275864);