Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393906 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2598 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
980 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,16)(8,13)(9,23)(10,14)(11,15)(12,19)(17,26)(18,33)(20,30)(21,28)(22,29)(24,27)(25,32)(31,34)(35,36);(1,2,3,4)(7,17,20,19)(12,16,26,30)(18,32,34,35)(25,31,36,33);(5,7)(6,12)(8,18)(9,17)(10,19)(11,20)(13,25)(14,30)(15,26)(16,23)(21,35)(22,34)(24,32)(27,31)(28,33)(29,36);(5,8,10,21,11,22,9,24)(6,13,14,28,15,29,23,27)(7,18,19,35,20,34,17,32)(12,25,30,33,26,36,16,31)Polycyclic presentation:
2^9|b^2=eh,d^2=gh,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,f^c=fi,g^c=gi,h^c=hi,f^d=fi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324447);