Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129558 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #418 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1824 |
| prime radical | C24 | C2×Q16 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
472 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 68 points:
(5,6,10,16,20,39,33,49,12,18,9,15,11,17,36,52)(7,19,23,42,37,50,31,13,25,41,22,53,24,47,34,43)(8,26,27,45,38,51,32,14,30,44,28,54,29,48,35,46)(21,56,57,68,62,65,60,66,59,40,63,64,58,67,61,55);(1,2)(3,4)(5,7,20,24,12,25,11,37)(6,13,39,42,18,43,17,53)(8,21,29,62,30,59,38,58)(9,31,36,23,10,34,33,22)(14,40,45,67,46,56,54,65)(15,47,52,41,16,50,49,19)(26,55,48,68,44,66,51,64)(27,61,35,57,28,60,32,63);(2,4)(5,8,20,29,12,30,11,38)(6,14,39,45,18,46,17,54)(7,21,24,62,25,59,37,58)(9,32,36,27,10,35,33,28)(13,40,42,67,43,56,53,65)(15,48,52,44,16,51,49,26)(19,55,47,68,41,66,50,64)(22,60,31,63,23,61,34,57)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=de,d^2=hi,e^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=ce,c^b=cf,d^b=dh,e^b=ehi,g^b=gh,h^b=hi,d^c=dhi,e^c=eh,g^c=gh,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56549);