Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12310 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1619 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #8977 |
| prime radical | #33594 | C2 |
| 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 | |
411 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 411 conjugacy classes of subgroups — too many to draw. Only its 94 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:
(2,7,11,16)(3,12,14,30,13,29,15,19)(9,23,26,25,24,27,21,32)(10,18,28,31);(1,2,8,7,6,11,4,16)(3,9,15,21,13,24,14,26)(5,18,22,28,20,31,17,10)(12,27,19,25,29,23,30,32);(1,3)(2,9)(4,14)(5,19)(6,13)(7,21)(8,15)(10,27)(11,24)(12,17)(16,26)(18,25)(20,30)(22,29)(23,28)(31,32)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=g,e^2=hi,f^2=gi,h^2=i,b^a=bd,c^a=ce,d^b=dg,e^b=eh,f^b=fg,h^b=hi,d^c=dh,e^c=eh,f^c=fh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72225390, #72225399, #72225403 |
| maximal normal subgroup of | #72225390, #72225399, #72225403 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44063);