Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149897 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×AΣL(1,𝔽16) |
| commutator | C22×D4 | C22×C4 |
| Frattini, cosocle | C2×#127 | C23 |
| prime radical | #24222 | C2 |
| LCS | G ▷ C22×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×#127 ◁ G | |
| LECS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
540 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 540 conjugacy classes of subgroups — too many to draw. Only its 66 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:
(2,4)(5,6)(7,10,15,19)(8,12,17,11)(9,13)(14,18,16,20);(5,7,9,14)(6,10,12,19,13,18,11,20)(8,16,17,15);(1,2,3,4)(7,14)(10,18)(15,16)(19,20)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,c^2=e,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,c^b=ci,d^b=dgi,f^b=fgi,g^b=gh,f^d=fhi,g^d=gi,g^f=gi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72279651 |
| maximal normal subgroup of | #72279651 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60104);