Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1339 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #327 |
| commutator | C2×C8 | C2×C16 |
| Frattini, cosocle | C2×#322 | C22 |
| socle | C22 | #1271 |
| prime radical | C23 | D4⋊C8 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×#322 ▷ C2×C42 ▷ C22 ▷ C1 | |
150 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 150 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 128 |
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)(17,18,19,24,22,26,30,23,20,27,29,31,21,25,28,32);(18,23,27,32)(19,28)(21,22)(24,25,31,26)(29,30)Polycyclic presentation:
2^9|a^2=d,b^2=eh,c^2=f,d^2=g,f^2=h,g^2=i,b^a=bc,c^a=ce,e^a=eh,c^b=cf,d^b=defh,f^b=fh,g^b=gh,d^c=dh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 70);