Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437452 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4920 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6)(7,11)(8,10)(9,13)(12,16)(14,20)(15,19)(17,18);(6,10)(8,16)(11,19)(12,13)(15,17)(18,20);(1,2,3,4)(5,7)(6,11)(8,17)(9,14)(10,18)(12,19)(13,20)(15,16);(8,16)(10,12)(15,17)(18,19);(7,14)(11,20)(15,17)(18,19)Polycyclic presentation:
2^9|c^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,f^b=fi,e^c=ei,f^c=fi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72219646, #72219652, #72219673, … |
| maximal normal subgroup of | #72219646, #72219652, #72219673, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6349761);