Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139718 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1898 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C23×Q8 | C23 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊Q8 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1 | |
666 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 666 conjugacy classes of subgroups — too many to draw. Only its 74 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:
(1,2,7,12,6,14,24,32)(3,5,18,4,19,21,29,20)(8,13,22,16,25,11,30,31)(9,23,26,15,17,27,10,28);(1,3,15,14,6,19,28,2)(4,11,25,17,20,13,8,9)(5,12,30,29,21,32,22,18)(7,26,23,16,24,10,27,31);(1,4,6,20)(2,9,14,17)(3,11,19,13)(5,22,21,30)(7,27,24,23)(8,15,25,28)(10,16,26,31)(12,18,32,29)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,c^2=i,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dgi,e^b=eh,f^b=fgi,g^b=gi,d^c=dh,f^c=fh,f^d=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72238412, #72238413, #72238416, … |
| maximal normal subgroup of | #72238412, #72238413, #72238416, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47029);