Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #119544 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1440 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
906 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 906 conjugacy classes of subgroups — too many to draw. Only its 118 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:
(1,2,4,11,5,7,18,22)(3,13,15,29,16,24,31,32)(6,12,20,30,21,10,19,28)(8,23,25,17,9,26,27,14);(2,7)(3,14)(4,18)(8,24)(9,13)(10,12)(15,26)(16,17)(19,20)(23,31)(25,29)(27,32);(2,8,7,9)(4,19)(10,13,12,24)(11,29,22,32)(15,23)(18,20)(25,30,27,28)(26,31)Polycyclic presentation:
2^9|a^2=g,c^2=d,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,e^b=ehi,g^b=gh,g^c=ghi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72233123, #72233126, #72233133, … |
| maximal normal subgroup of | #72233123, #72233126, #72233133, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 26741);