Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573652 |
| 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 | #60376 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2596 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,18)(12,13)(14,17)(15,31)(16,20)(19,28)(21,37)(22,26)(23,24)(25,33)(27,29)(30,32)(34,35)(36,38)(39,40);(1,3,5,4)(2,6,7,8)(9,11,17,25)(10,18,14,33)(12,26,30,39)(13,22,32,40)(15,24,16,34)(19,36,27,37)(20,35,31,23)(21,28,38,29);(9,12)(10,19)(11,21)(13,31)(14,27)(15,29)(16,28)(17,30)(18,22)(20,32)(23,37)(24,39)(25,38)(26,34)(33,40)(35,36);(1,4,5,3)(2,6,7,8)(9,13,17,32)(10,12,14,30)(11,22,25,40)(15,27,16,19)(18,26,33,39)(20,28,31,29)(21,35,38,23)(24,37,34,36);(9,14)(10,17)(11,23)(12,27)(13,29)(15,31)(16,20)(18,24)(19,30)(21,37)(22,39)(25,35)(26,40)(28,32)(33,34)(36,38)Polycyclic presentation:
2^9|b^2=fi,d^2=fi,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,g^c=gi,h^c=hi,e^d=eh,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6483936);