Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9331 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
828 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6)(7,13,15,10)(8,11,16,18)(9,12)(14,17)(19,20)(21,22,26,29,27,30,25,24)(23,31,32,28,33,36,34,35);(1,2,3,4)(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(21,23)(22,28)(24,31)(25,34)(26,32)(27,33)(29,36)(30,35);(5,8)(6,11,12,18)(7,14)(9,16)(10,17,13,19)(15,20)(21,24,26,30,27,29,25,22)(23,31,32,35,33,36,34,28)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,d^b=di,e^b=ehi,f^b=fhi,d^c=dhi,e^c=eh,f^c=fh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 17930);