Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #397240 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#394 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1497 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)(5,6,11,14,9,15,10,24)(7,16,18,28,19,12,25,31)(8,20,26,30,22,34,23,13)(17,32,33,29,21,35,36,27);(5,7,11,25,9,19,10,18)(6,12,24,31,15,16,14,28)(8,21,26,33,22,17,23,36)(13,29,34,35,30,27,20,32);(5,8,9,22)(6,13)(7,17)(10,23,11,26)(12,27,16,29)(14,20)(15,30)(18,33)(19,21)(24,34)(25,36)(28,35,31,32);(2,4)Polycyclic presentation:
2^9|a^2=ef,b^2=fi,c^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 304437);