Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139572 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1910 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | C4×M16 | C23 |
| socle | C2 | #9492 |
| prime radical | #2010 | C4 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C4×M16 ◁ G | |
| LECS | G ▷ C4×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
344 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 344 conjugacy classes of subgroups — too many to draw. Only its 64 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:
(1,2,5,11,6,22,4,19,7,9,21,10,14,12,20,26)(3,13,17,25,18,32,16,8,15,24,31,27,29,28,30,23);(1,3,14,29,7,15,6,18)(2,8,9,23)(4,16,5,17,20,30,21,31)(10,13,11,24)(12,25,22,27)(19,28,26,32);(2,9)(3,15)(12,22)(13,24)(16,30)(17,31)(18,29)(28,32)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=gi,f^2=gh,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,c^b=ci,e^b=ehi,d^c=dh,f^c=fh,e^d=ei,f^e=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72245146, #72245189, #72306781, … |
| maximal normal subgroup of | #72245189, #72306781, #72306788, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46891);