Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94626 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7394 |
| commutator | #379 | C2×C4 |
| Frattini, cosocle | #1781 | C22 |
| prime radical | #10336 | C2 |
| LCS | G ▷ #379 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C24 ◁ C25 ◁ #1781 ◁ G | |
| LECS | G ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
445 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 445 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) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,4,7,5,8,13,16)(3,9,10,15,11,6,12,14);(2,6)(7,15)(8,9)(10,12)(14,16)Polycyclic presentation:
2^9|a^2=d,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 | #72236933, #72283320, #72283335, … |
| maximal normal subgroup of | #72236933, #72283320, #72283335, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1824);