Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573699 |
| 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 | |
4453 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)(4,9)(5,21)(6,10)(7,11)(8,13)(14,27)(15,16)(17,24)(18,25)(19,22)(20,26)(23,28)(29,30)(31,32);(1,3)(2,8)(4,14)(5,17)(6,13)(7,16)(9,23)(10,15)(11,24)(12,21)(18,31)(19,30)(20,29)(22,28)(25,27)(26,32);(3,13)(4,18)(5,6)(8,15)(9,25)(10,21)(12,24)(14,29)(16,17)(19,26)(20,22)(23,32);(1,4,7,20)(2,9,11,26)(3,14,16,29)(5,18,6,22)(8,23,24,32)(10,19,21,25)(12,27,15,30)(13,28,17,31);(1,2)(3,15)(4,19)(5,10)(6,21)(7,11)(8,13)(9,22)(12,16)(14,23)(17,24)(18,26)(20,25)(27,28)(29,32)(30,31)Polycyclic presentation:
2^9|d^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,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=hi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72215710, #72215784, #72237523, … |
| maximal normal subgroup of | #72215784, #72237523, #72312363, … |
| maximal quotient of | #72312219, #72312221, #72312223, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6480896);