Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395552 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#394 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2190 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)(5,6,9,25,11,15,10,14)(7,16,18,12,20,28,19,29)(8,21,26,13,24,31,23,32)(17,27,22,30,34,36,33,35);(5,7,9,18,11,20,10,19)(6,12,14,16,15,29,25,28)(8,22,23,17,24,33,26,34)(13,30,31,36,32,35,21,27);(5,8)(6,13)(7,17)(9,23)(10,26)(11,24)(12,27)(14,31)(15,32)(16,30)(18,33)(19,22)(20,34)(21,25)(28,35)(29,36);(2,4)Polycyclic presentation:
2^9|a^2=fi,b^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302749);