Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143203 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1619 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
575 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 575 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(6,10)(7,14)(8,17)(11,19)(12,16)(13,20)(15,18);(1,2)(3,4)(6,11)(7,15)(8,17)(12,13)(14,18)(16,19);(2,4)(5,6)(7,16)(8,12)(9,13)(10,14)(11,17)(15,20)(18,19)Polycyclic presentation:
2^9|d^2=g,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=dg,e^b=ehi,f^b=fi,h^b=hi,d^c=dh,e^c=eh,f^c=fi,g^c=gi,h^c=hi50 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#143203, #72214466, #72214645, … |
| maximal normal subgroup of | C2×#143203, #72214466, #72214645, … |
| maximal quotient of | #72213160, C2×#143203, #72215015, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50399);