Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139432 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1898 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
644 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 644 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:
(1,2,6,13,7,12,5,26)(3,16,20,9,19,30,28,25)(4,21,23,32,24,10,22,18)(8,14,29,31,17,15,27,11);(1,3,17,10)(2,9,15,4)(5,20,6,28)(7,19,8,21)(11,30,31,16)(12,25,14,24)(13,23,26,22)(18,27,32,29);(1,4,17,9)(2,10,15,3)(5,23,29,30)(6,22,27,16)(7,24,8,25)(11,28,13,32)(12,21,14,19)(18,31,20,26)Polycyclic presentation:
2^9|a^2=f,b^2=dfhi,c^2=ghi,d^2=ghi,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gi,h^a=hi,d^b=dh,e^b=egh,f^b=fhi,g^b=gi,h^b=hi,d^c=dgh,f^c=fgh,f^d=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72238635, #72238638, #72238642, … |
| maximal normal subgroup of | #72238635, #72238638, #72238642, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46719);