Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12310 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1619 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #8920 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
350 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 350 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,11,7,4,18,10)(3,12,16,27,15,30,17,31)(5,20,23,9,24,22,28,19)(8,25,26,32,13,21,29,14);(1,3,6,16,7,15,18,17)(2,8,11,26,4,13,10,29)(5,21,23,14,24,25,28,32)(9,27,22,30,19,31,20,12);(1,4)(2,7)(3,13)(5,22)(6,10)(8,15)(9,28)(11,18)(12,25)(14,31)(16,29)(17,26)(19,23)(20,24)(21,30)(27,32)Polycyclic presentation:
2^9|a^2=f,b^2=f,d^2=gi,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dg,e^b=eg,g^b=gi,h^b=hi,d^c=dg,e^c=eh,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72225390, #72225403 |
| maximal normal subgroup of | #72225390, #72225403 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43686);