Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6629032 |
| 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 | |
4573 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)(6,10)(7,13)(8,16)(12,18)(14,15)(17,20);(5,6,8,10,9,12,16,18)(7,11,14,17,15,19,13,20);(5,7)(6,11)(8,14)(9,15)(10,17)(12,19)(13,16)(18,20);(2,4);(6,12)(10,18)(11,19)(17,20)Polycyclic presentation:
2^9|b^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,e^b=ei,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72238735 |
| maximal normal subgroup of | #72238735 |
| maximal quotient of | #72238755, #72321838, #72323036 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536228);