Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94626 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7394 |
| commutator | #379 | C2×C4 |
| Frattini, cosocle | #1781 | C22 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ #379 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C24 ◁ C25 ◁ #1781 ◁ G | |
| LECS | G ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
303 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 303 conjugacy classes of subgroups — too many to draw. Only its 22 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 — 2 generators on 32 points:
(1,2,5,9,6,10,19,25)(3,11,13,26,14,27,30,32)(4,15,16,31,17,8,18,23)(7,20,21,12,22,28,24,29);(1,3,6,14)(2,7,10,22)(4,12,17,29)(5,13,19,30)(8,11,15,27)(9,24,25,21)(16,20,18,28)(23,32,31,26)Polycyclic presentation:
2^9|a^2=d,b^2=i,c^2=f,d^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fhi,g^a=gi,h^a=hi,c^b=cf,d^b=def,g^b=gh,d^c=dg,e^c=ehi,g^c=gi,e^d=ei,f^d=fi29 conjugacy classes — view the full character table.
| maximal subgroup of | #72283443, #72283503, #72283527, … |
| maximal normal subgroup of | #72283443, #72283503, #72283527, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1825);