Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12483 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #459 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
362 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 362 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6)(7,15)(8,11)(9,18)(10,14)(12,16)(13,19)(17,20)(21,22,24,29,33,36,32,27,26,31,23,28,25,30,34,35);(1,2,3,4)(5,7,9,8,10,16,19,17)(6,11,13,15,14,20,18,12)(22,27)(23,32)(24,34)(25,33)(28,30)(29,36)(31,35);(5,8,9,16,10,17,19,7)(6,12,13,11,14,15,18,20)Polycyclic presentation:
2^9|a^2=f,b^2=dfhi,c^2=e,d^2=ghi,e^2=g,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,c^b=cg,d^b=dh,f^b=fh,h^b=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48480);