Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396373 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2416 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1038 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(2,3)(5,6,11,16,12,17,10,31)(7,18,32,36,22,37,21,13)(8,14,25,41,26,42,33,40)(9,27,29,43,30,56,28,15)(19,38,46,34,47,61,23,51)(20,48,57,60,50,35,49,44)(24,53,54,39,55,63,58,64)(45,65,66,62,67,59,52,68);(1,2,4,3)(5,7,10,21,12,22,11,32)(6,13,16,18,17,36,31,37)(8,23,25,19,26,46,33,47)(9,20,28,49,30,50,29,57)(14,38,40,51,42,61,41,34)(15,35,27,44,43,48,56,60)(24,52,54,45,55,66,58,67)(39,62,53,65,64,68,63,59);(1,3,4,2)(5,8,10,25,12,26,11,33)(6,14,16,40,17,42,31,41)(7,19,21,46,22,47,32,23)(9,24,28,54,30,55,29,58)(13,34,18,38,36,51,37,61)(15,39,27,53,43,64,56,63)(20,45,49,66,50,67,57,52)(35,59,44,62,48,65,60,68);(5,9,11,29,12,30,10,28)(6,15,31,56,17,43,16,27)(7,20,32,57,22,50,21,49)(8,24,33,58,26,55,25,54)(13,35,37,60,36,48,18,44)(14,39,41,63,42,64,40,53)(19,45,23,52,47,67,46,66)(34,59,61,68,51,65,38,62)Polycyclic presentation:
2^9|a^2=efi,b^2=e,c^2=e,d^2=hi,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,g^b=gi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 319935);