Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573451 |
| 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 | #2388 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1967 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)(7,11)(8,10)(9,12)(14,19)(16,28)(20,34)(22,23)(24,39)(25,42)(29,43)(30,31);(5,7,9,8)(6,10,12,11)(13,14)(15,24)(16,21)(17,19)(18,22)(20,27)(23,32)(25,41)(26,39)(28,35)(29,44)(30,38)(31,40)(33,34)(36,43)(37,42);(2,4)(5,7,9,8)(6,11,12,10)(13,15,17,26)(14,20,19,34)(16,29,28,43)(18,33,32,27)(21,37,35,41)(22,24,23,39)(25,30,42,31)(36,40,44,38);(5,7,9,8)(6,11,12,10)(13,16,17,28)(14,21,19,35)(15,25,26,42)(18,30,32,31)(20,36,34,44)(22,38,23,40)(24,41,39,37)(27,43,33,29);(5,8,9,7)(6,10,12,11)(14,22)(16,30)(18,32)(19,23)(20,24)(25,43)(27,33)(28,31)(29,42)(34,39)(36,44)(38,40)Polycyclic presentation:
2^9|b^2=fi,c^2=f,d^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6486245);