Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #112268 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C2×#2661 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1215 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 1215 conjugacy classes of subgroups — too many to draw. Only its 134 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:
(1,2,4,11,6,12,19,29)(3,13,16,28,14,22,31,32)(5,10,18,27,20,8,17,24)(7,21,23,15,9,25,26,30);(2,7)(3,14)(4,17)(8,13)(9,12)(10,22)(11,28)(15,30)(16,25)(18,19)(21,31)(23,24)(26,27)(29,32);(2,8)(3,15)(4,17)(7,9)(10,12)(13,22)(14,30)(16,31)(18,19)(21,25)(23,32)(26,28)Polycyclic presentation:
2^9|a^2=g,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,c^b=cf,d^b=di,g^b=ghi,d^c=dh,g^c=ghi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72237982, #72237987, #72237989, … |
| maximal normal subgroup of | #72237982, #72237987, #72237989, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 19470);