Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9103 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1436 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1442 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6)(7,17,20,12)(8,21,23,30)(9,15)(10,16)(11,14)(13,22,29,24)(18,32)(19,31,25,27)(26,33)(28,36)(34,35);(1,2)(3,4)(5,7,10,19,11,20,9,25)(6,12,16,27,14,17,15,31)(8,18,24,33,23,34,22,36)(13,28,30,32,29,26,21,35);(2,4)(5,8)(6,13,14,29)(7,18)(9,22)(10,24)(11,23)(12,26,17,28)(15,21,16,30)(19,33)(20,34)(25,36)(27,35,31,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=d,e^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,e^b=ei,g^b=gi,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30905);