Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143695 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10228 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
456 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 456 conjugacy classes of subgroups — too many to draw. Only its 78 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 — 3 generators on 32 points:
(2,9)(3,15)(4,19)(5,22)(6,7)(10,27)(11,14)(12,20)(13,26)(17,31)(18,21)(23,24)(25,29)(28,32);(1,2)(3,16)(4,20)(5,11)(6,13)(7,12)(8,14)(9,19)(10,17)(15,25)(18,30)(21,28)(22,26)(23,29)(24,27)(31,32);(1,3)(2,10)(4,21)(5,23)(6,24)(7,17)(8,18)(9,25)(11,28)(12,30)(13,16)(14,27)(15,22)(19,31)(20,32)(26,29)Polycyclic presentation:
2^9|d^2=gi,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,f^b=fi,g^b=gi,d^c=di,e^c=eh,f^c=fi,h^c=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50891);