Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511896 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #33175 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
673 conjugacy classes of subgroups; only the 156 normal subgroups are shown.
This group has 673 conjugacy classes of subgroups — too many to draw. Only its 156 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) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,3)(6,12)(7,15)(8,10)(9,13)(14,17)(16,19)(18,20);(1,2,4,3)(5,6,8,12,11,13,10,9)(7,14,18,20,17,15,16,19);(6,13)(9,12)(14,15)(19,20);(1,3,4,2)(5,7)(6,14)(8,18)(9,19)(10,16)(11,17)(12,20)(13,15)Polycyclic presentation:
2^9|b^2=ef,d^2=eh,e^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gh,h^a=hi,c^b=ci,g^b=gi,g^c=gi,e^d=ei,f^d=fi,g^d=gh,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#512308, #72226767 |
| maximal normal subgroup of | C2×#512308, #72226767 |
| maximal quotient of | C2×#512308 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419504);