Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7339 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1316 |
| commutator | C22×C4 | C2×C16 |
| Frattini, cosocle | #1817 | C22 |
| prime radical | C23×D4 | C8 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #1817 ◁ G | |
| LECS | G ▷ #1817 ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
401 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 401 conjugacy classes of subgroups — too many to draw. Only its 34 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) = 64 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,3,6,4,7,9,12,5,8,10,13,11,14,15,16)(17,18,20,25,22,23,29,30,21,26,28,32,19,24,27,31);(18,23)(19,22)(20,28)(24,26)(25,30)(31,32)Polycyclic presentation:
2^9|a^2=d,c^2=i,d^2=f,f^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gi,c^b=ci,d^b=dei,f^b=fi,d^c=dg,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 869);