Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13392 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #482 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #2077 |
| prime radical | C24 | #137 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 68 points:
(5,6,11,18,12,20,36,19)(7,21,26,50,27,52,37,39)(8,28,32,46,33,45,66,61)(9,17,34,15,35,51,10,16)(13,25,40,23,42,59,41,24)(14,43,47,65,49,31,48,30)(22,54,57,38,58,60,63,67)(29,62,55,68,64,44,56,53);(1,2)(3,4)(5,7,11,37,12,27,36,26)(6,13,18,41,20,42,19,40)(8,29,32,56,33,64,66,55)(9,25,34,24,35,59,10,23)(14,44,47,68,49,62,48,53)(15,50,51,21,16,39,17,52)(22,30,63,43,58,65,57,31)(28,60,46,38,45,54,61,67);(2,4)(5,8,9,30,11,32,34,43,12,33,35,65,36,66,10,31)(6,14,15,45,19,48,17,46,20,49,16,28,18,47,51,61)(7,22,23,55,26,57,59,64,27,58,24,56,37,63,25,29)(13,38,21,53,41,54,39,44,42,67,52,68,40,60,50,62)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=d,d^2=h,e^2=h,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,h^a=hi,c^b=cf,d^b=dhi,e^b=eh,f^b=fh,g^b=gi,h^b=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56335);