Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6431723 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
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 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
7590 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(3,4)(5,6)(7,11)(8,10)(9,13)(12,16)(14,18)(15,17)(19,20);(6,10)(8,16)(11,17)(12,13)(15,20)(18,19);(2,4)(8,16)(10,12)(15,20)(17,19);(5,7)(6,11)(8,15)(9,14)(10,17)(12,19)(13,18)(16,20);(7,14)(11,18)(15,20)(17,19)Polycyclic presentation:
2^9|f^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,f^b=fi,f^c=fi,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72215901, #72215961, #72215974, … |
| maximal normal subgroup of | #72215901, #72215961, #72215974, … |
| maximal quotient of | C2×#6431723, #72261762, #72261773 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6338919);