Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437529 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3224 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,25)(16,21)(17,24)(18,23)(19,22)(20,28)(26,37)(27,36)(29,35)(30,34)(31,40)(32,39)(33,38)(41,44)(42,43);(2,4)(13,15,19,29)(14,20,22,36)(16,30,33,41)(17,27,18,28)(21,37,38,43)(23,35,24,25)(26,31,42,32)(34,39,44,40);(6,9)(10,12)(13,16,19,33)(14,21,22,38)(15,26,29,42)(17,32,18,31)(20,34,36,44)(23,40,24,39)(25,37,35,43)(27,41,28,30);(5,7,8,11)(6,10,9,12)(15,27)(17,18)(20,35)(23,24)(25,36)(26,41)(28,29)(30,42)(31,32)(34,43)(37,44)(39,40);(14,22)(20,36)(21,38)(23,24)(25,35)(34,44)(37,43)(39,40)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,h^a=hi,c^b=ch,d^b=dhi,f^b=fi,h^b=hi,f^c=fi,h^c=hi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6356009);