Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7540 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1415 |
| commutator | C32 | C42 |
| Frattini, cosocle | C22×C32 | C22 |
| prime radical | C23 | SD64 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ C22×C32 ◁ G | |
| LECS | G ▷ C22×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
106 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 68 points:
(1,2,3,4)(6,13)(7,19)(8,25)(9,21)(10,31)(11,33)(14,40)(15,44)(16,47)(17,49)(18,20)(22,28)(23,30)(24,29)(26,57)(27,61)(32,58)(34,60)(35,37)(36,39)(38,45)(41,67)(42,46)(43,66)(48,68)(50,53)(51,52)(54,64)(55,56)(59,62)(63,65);(5,6,8,14,26,41,22,54,31,45,62,44,58,66,7,20,11,17,29,37,55,63,51,68,34,48,61,65,21,35,23,49,12,18,30,43,9,15,27,38,60,64,52,67,56,40,24,13,33,36,19,50,32,46,59,47,10,16,28,42,57,53,25,39)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fg,e^2=fi,f^2=gh,g^2=hi,h^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fg,g^a=gh,h^a=hi,c^b=ci92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1964);