Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12026 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #427 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1808 |
| prime radical | C24 | C2×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×M16 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
536 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 68 points:
(2,4)(5,6)(7,19,23,44)(8,25)(9,16)(10,32)(11,34)(12,18)(13,21,37,50)(14,27)(15,30)(17,33)(20,45,48,35)(22,52,51,38)(24,54,53,36)(26,56)(28,39)(29,58)(31,42)(40,60)(41,59)(43,63,64,47)(46,65,67,55)(49,62,68,66)(57,61);(5,7,10,22,33,53,31,50,12,23,30,51,11,24,9,21)(6,13,16,36,34,52,15,19,18,37,42,54,17,38,32,44)(8,20,27,47,61,68,60,67,29,48,59,63,28,49,26,46)(14,35,25,55,56,66,39,43,41,45,58,65,40,62,57,64);(1,2,3,4)(5,8,9,26,11,28,30,59,12,29,31,60,33,61,10,27)(6,14,32,57,17,40,42,58,18,41,15,39,34,56,16,25)(7,20,21,46,24,49,51,63,23,48,50,67,53,68,22,47)(13,35,44,64,38,62,54,65,37,45,19,43,52,66,36,55)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=eh,d^2=h,e^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,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, 54292);