Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1412 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #354 |
| commutator | C42 | C2×C16 |
| Frattini, cosocle | #2150 | C22 |
| socle | C2 | #7057 |
| prime radical | C23 | #349 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×C8 ◁ C22×C8 ◁ #2150 ◁ G | |
| LECS | G ▷ #2150 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
109 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,4,10,6,12,22,37,5,11,24,36,25,39,53,60,7,13,23,38,27,40,52,61,26,9,21,33,50,34,51,58)(3,14,16,41,18,42,45,62,19,8,28,29,47,30,54,55,15,31,43,35,44,56,63,59,17,32,20,48,46,57,49,64);(1,3,7,15)(2,8)(4,20,23,28)(5,19,26,17)(6,18,27,44)(9,14)(10,35)(11,31)(12,30)(13,32)(16,21,43,24)(22,49,52,54)(25,47,50,46)(29,33)(34,42)(36,48)(37,59)(38,41)(39,56)(40,57)(45,51,63,53)(55,58)(60,64)(61,62)Polycyclic presentation:
2^9|a^2=d,b^2=g,c^2=gi,d^2=f,e^2=i,f^2=h,h^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gi,c^b=cg,d^b=deg,e^b=ei,d^c=dgi,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 895);