Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498795 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | C2×D8 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
822 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 44 points:
(5,6,8,10)(7,11,12,9)(13,14,16,22,30,37,41,33,18,21,31,34,17,23,28,19)(15,24,25,39,32,44,43,35,27,40,42,36,26,38,29,20);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(14,19)(16,28)(17,30)(20,24)(21,33)(22,23)(25,29)(26,32)(31,41)(34,37)(35,40)(36,44)(38,39)(42,43);(2,4)(13,15,18,27)(14,20,21,35)(16,29,31,43)(17,32,30,26)(19,24,33,40)(22,38,34,44)(23,39,37,36)(25,41,42,28);(14,21)(19,33)(20,35)(22,34)(23,37)(24,40)(36,39)(38,44)Polycyclic presentation:
2^9|a^2=e,b^2=efh,c^2=i,e^2=hi,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,e^b=eh,f^b=fh,h^b=hi,e^c=eh,f^c=fh,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406725);