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:
(5,6,12,19,13,21,37,20)(7,22,26,42,28,55,27,52)(8,29,34,62,35,47,65,48)(9,36,11,16,10,18,39,17)(14,25,43,40,44,38,53,24)(15,45,49,31,51,33,50,32)(23,56,58,41,60,61,59,66)(30,46,63,67,64,68,57,54);(1,2)(3,4)(5,7,12,26,13,28,37,27)(6,14,20,53,21,44,19,43)(8,30,34,63,35,64,65,57)(9,25,11,40,10,38,39,24)(15,46,50,54,51,68,49,67)(16,52,36,55,17,42,18,22)(23,32,58,45,60,31,59,33)(29,61,48,41,47,56,62,66);(2,4)(5,8,11,33,12,34,10,32,13,35,39,45,37,65,9,31)(6,15,36,62,20,50,17,29,21,51,18,48,19,49,16,47)(7,23,38,64,27,59,40,63,28,60,25,30,26,58,24,57)(14,41,42,46,43,61,55,67,44,66,52,68,53,56,22,54)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=dh,d^2=hi,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,h^a=hi,c^b=cf,d^b=dhi,e^b=eh,f^b=fhi,h^b=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56361);