Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437793 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4678 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(3,4)(5,6,9,15,11,16,10,14)(7,12,18,17,8,13,20,19);(5,6)(7,17)(8,19)(9,14)(10,15)(11,16)(12,18)(13,20);(2,4)(5,7)(6,12)(8,11)(9,18)(10,20)(13,16)(14,19)(15,17);(5,8)(6,13)(7,11)(9,20)(10,18)(12,16)(14,17)(15,19);(5,9,11,10)(6,14,16,15)(7,18,8,20)(12,19,13,17)Polycyclic presentation:
2^9|a^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72238734 |
| maximal normal subgroup of | #72238734 |
| maximal quotient of | #72238755 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344992);