Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12026 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #427 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1835 |
| prime radical | C24 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#117 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
484 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)(7,20,23,36)(8,26)(9,17)(10,16)(11,34)(12,19)(13,22,37,52)(14,27)(15,31)(18,33)(21,47,49,64)(24,46,53,39)(25,55,54,38)(28,42)(29,41)(30,40)(32,44)(35,51,62,68)(43,60)(45,65,56,48)(50,63,67,66)(57,59)(58,61);(5,7,10,24,33,54,32,52,12,23,31,53,11,25,9,22)(6,13,17,38,34,46,15,20,19,37,44,55,18,39,16,36)(8,21,28,50,61,68,60,65,30,49,59,67,29,51,27,48)(14,35,41,63,57,47,40,45,43,62,58,66,42,64,26,56);(2,4)(5,8,9,27,11,29,31,59,12,30,32,60,33,61,10,28)(6,14,15,40,18,42,17,41,19,43,16,26,34,57,44,58)(7,21,22,48,25,51,53,67,23,49,52,65,54,68,24,50)(13,35,20,45,39,64,38,63,37,62,36,56,46,47,55,66)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=d,d^2=h,e^2=h,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=gh,h^a=hi,f^b=fi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54123);