Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13312 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #469 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #2034 |
| prime radical | C24 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 68 points:
(1,2,3,4)(5,6,12,18,13,19,35,49,10,17,11,36,21,39,9,16)(7,20,24,44,25,41,23,50,26,52,34,14,38,42,37,43)(8,27,32,60,33,61,54,46,30,15,31,47,53,48,29,59)(22,51,28,62,56,45,55,67,57,68,63,40,64,58,65,66);(5,7,21,25,10,26,13,38)(6,14,39,43,17,44,19,50)(8,28,53,55,30,63,33,65)(9,34,11,37,35,24,12,23)(15,45,61,68,27,58,48,51)(16,41,36,52,49,42,18,20)(22,54,56,32,57,29,64,31)(40,46,66,60,62,59,67,47);(5,8,9,29,21,53,11,31,10,30,35,54,13,33,12,32)(6,15,16,46,39,61,36,60,17,27,49,59,19,48,18,47)(7,22,23,55,38,64,24,28,26,57,37,65,25,56,34,63)(14,40,20,51,50,67,42,58,44,62,52,68,43,66,41,45)Polycyclic presentation:
2^9|a^2=g,b^2=eh,c^2=d,d^2=hi,f^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,d^b=dh,e^b=ei,f^b=fhi,g^b=gh,h^b=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56267);