Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419810 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23387 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1261 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)(5,6,9,13,11,16,10,15)(7,17,25,28,18,27,20,12)(8,14,23,31,24,32,22,29)(19,34,21,35,33,30,36,26);(5,7,10,20,11,18,9,25)(6,12,15,27,16,28,13,17)(8,21,23,33,24,36,22,19)(14,30,31,26,32,34,29,35);(6,13)(7,18)(9,10)(12,17)(14,29)(15,16)(19,33)(22,23)(26,30)(27,28)(31,32)(34,35);(2,4)(5,8)(6,14,16,32)(7,19,18,33)(9,22)(10,23)(11,24)(12,26)(13,29,15,31)(17,30)(20,21,25,36)(27,34)(28,35)Polycyclic presentation:
2^9|a^2=eg,b^2=f,d^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 343902);