Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573643 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #60372 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2687 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,17)(12,19)(13,20)(14,18)(15,21)(16,22)(23,31)(24,32)(25,33)(26,35)(27,38)(28,37)(29,34)(30,36)(39,40);(1,3,5,4)(2,6,7,8)(9,11,16,25)(10,17,22,33)(12,14,24,15)(13,26,27,39)(18,32,21,19)(20,35,38,40)(23,30,28,29)(31,36,37,34);(10,18)(11,12)(13,27)(14,15)(19,32)(20,36)(21,22)(23,39)(24,25)(26,28)(31,37)(34,38);(1,4,5,3)(2,6,7,8)(9,12)(10,19)(11,14)(13,28)(15,25)(16,24)(17,18)(20,37)(21,33)(22,32)(23,27)(26,29)(30,39)(31,38)(34,35)(36,40);(9,13)(10,20)(11,23)(12,26)(14,29)(15,30)(16,27)(17,31)(18,34)(19,35)(21,36)(22,38)(24,39)(25,28)(32,40)(33,37)Polycyclic presentation:
2^9|b^2=fi,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,h^b=hi,d^c=dh,e^c=ei,g^c=gi,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, 6483944);