Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437418 |
| 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 | C2×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2000 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,10)(8,9)(11,12)(13,14)(15,25)(16,21)(17,22)(18,33)(19,24)(20,28)(23,30)(26,36)(27,41)(29,40)(31,38)(32,39)(34,37)(35,43)(42,44);(2,3)(13,15,19,29)(14,20,24,37)(16,26,31,42)(17,27,32,43)(18,34,30,28)(21,35,38,41)(22,36,39,44)(23,40,33,25);(1,2,4,3)(6,9)(10,12)(13,16)(14,21)(15,26)(17,18)(19,31)(20,35)(22,33)(23,39)(24,38)(25,36)(27,34)(28,43)(29,42)(30,32)(37,41)(40,44);(1,3,4,2)(13,17)(14,22)(15,27)(16,30)(18,31)(19,32)(20,36)(21,23)(24,39)(25,41)(26,28)(29,43)(33,38)(34,42)(35,40)(37,44);(5,7,8,11)(6,10,9,12)(16,31)(17,32)(21,38)(22,39)(26,42)(27,43)(35,41)(36,44)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6374421);