Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573703 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | D4×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4429 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)(3,12)(4,18)(5,22)(6,10)(7,11)(8,13)(9,21)(14,29)(15,16)(17,25)(19,20)(23,26)(24,31)(27,30)(28,32);(1,3)(2,8)(4,14)(5,17)(6,13)(7,16)(9,24)(10,15)(11,25)(12,22)(18,32)(19,31)(20,29)(21,30)(23,28)(26,27);(3,13)(4,19)(5,6)(8,15)(9,26)(10,22)(12,25)(14,30)(16,17)(18,20)(21,23)(24,32);(1,4)(2,9)(3,14)(5,19)(6,23)(7,21)(8,24)(10,20)(11,18)(12,27)(13,28)(15,29)(16,30)(17,31)(22,26)(25,32);(1,2)(3,15)(4,20)(5,10)(6,22)(7,11)(8,13)(9,23)(12,16)(14,24)(17,25)(18,19)(21,26)(27,28)(29,31)(30,32)Polycyclic presentation:
2^9|f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,e^b=eh,f^b=fi,g^b=gi,h^b=hi,d^c=dh,f^c=fi,h^c=hi,e^d=eh,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72215654, #72215784, #72236997, … |
| maximal normal subgroup of | #72215784, #72236997, #72237525, … |
| maximal quotient of | #72227326, #72311981, #72312189, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6480899);