Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149745 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×AΣL(1,𝔽16) |
| commutator | C22×D4 | C22×C4 |
| Frattini, cosocle | C2×#124 | C23 |
| prime radical | C2×#3018 | C2 |
| LCS | G ▷ C22×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1355 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 1355 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) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6,9,14,11,7,15,18)(8,12,19,13,10,16,20,17);(5,7)(6,11)(8,16)(9,17)(10,12)(13,15)(14,19)(18,20);(2,4)(5,8,11,10)(6,12,7,16)(9,19,15,20)(13,18,17,14)Polycyclic presentation:
2^9|a^2=f,c^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,f^b=fgi,g^b=gi,h^b=hi,f^d=fhi,g^d=gi,g^f=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72227997, #72228004, #72228006, … |
| maximal normal subgroup of | #72227997, #72228004, #72228006, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59208);