Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143162 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10018 |
| commutator, Frattini, cosocle | C4×M16 | C23 |
| prime radical | #30057 | C2 |
| LCS | G ▷ C4×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ G | |
| LECS | G ▷ C4×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
379 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 379 conjugacy classes of subgroups — too many to draw. Only its 50 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:
(1,2,7,12)(3,13,17,31)(4,18,21,28)(5,11,6,10)(8,23,26,14)(9,20,29,22)(15,27,16,30)(19,24,32,25);(1,3)(2,8)(4,19)(5,15)(6,16)(7,17)(9,27)(10,24)(11,25)(12,26)(13,18)(14,22)(20,23)(21,32)(28,31)(29,30);(4,20)(5,6)(9,28)(10,11)(13,27)(14,19)(15,16)(18,29)(21,22)(23,32)(24,25)(30,31)Polycyclic presentation:
2^9|a^2=i,d^2=gi,e^2=gh,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fgh,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eg,g^b=gi,d^c=dh,e^c=egh,g^c=gi,h^c=hi,f^d=fi35 conjugacy classes — view the full character table.
| maximal subgroup of | #72245149, #72245189, #72306550, … |
| maximal normal subgroup of | #72245189, #72306550, #72306767, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50360);