Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574112 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | D4×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2474 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,15)(4,19)(5,9)(6,12)(7,14)(8,13)(10,17)(11,22)(16,31)(18,25)(20,30)(21,27)(23,26)(24,29)(28,32);(1,3,7,17,9,6,8,18)(2,10,13,15,5,25,14,12)(4,20,22,21,24,32,23,16)(11,28,19,27,26,30,29,31);(1,4)(2,11)(3,16)(5,26)(6,21)(7,22)(8,23)(9,24)(10,27)(12,28)(13,19)(14,29)(15,30)(17,20)(18,32)(25,31);(1,5)(2,9)(3,10)(4,19)(6,25)(7,14)(8,13)(11,23)(12,18)(15,17)(16,30)(20,31)(21,28)(22,26)(24,29)(27,32);(1,6)(2,12)(3,9)(4,21)(5,15)(7,18)(8,17)(10,13)(11,28)(14,25)(16,24)(19,27)(20,23)(22,32)(26,30)(29,31)Polycyclic presentation:
2^9|b^2=f,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dh,g^c=gi,h^c=hi,e^d=eh,h^d=hi,h^e=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481308);