Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2709 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)(15,19)(16,20)(17,22)(18,21)(23,30)(24,34)(25,36)(26,35)(27,37)(28,39)(29,38)(31,33)(32,40)(41,43)(42,44);(6,9)(10,12)(13,15,18,26)(14,19,21,35)(16,27,29,42)(17,30,33,25)(20,37,38,44)(22,23,31,36)(24,32,41,28)(34,40,43,39);(2,4)(5,7,8,11)(6,10,9,12)(13,14)(15,23)(16,20)(17,31)(18,21)(19,30)(22,33)(24,37)(25,35)(26,36)(27,34)(28,40)(29,38)(32,39)(41,44)(42,43);(13,16,18,29)(14,20,21,38)(15,24,26,41)(17,32,33,28)(19,34,35,43)(22,40,31,39)(23,37,36,44)(25,42,30,27);(14,21)(19,35)(20,38)(22,31)(23,36)(34,43)(37,44)(39,40)Polycyclic presentation:
2^9|b^2=i,c^2=f,d^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345439);