Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1624 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #10228 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
439 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 439 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:
(1,2)(3,16)(4,21)(5,26)(6,29)(7,13)(8,15)(9,17)(10,23)(11,28)(12,31)(14,27)(18,25)(19,32)(20,22)(24,30);(1,3)(2,9)(4,15)(5,18)(6,30)(7,19)(8,20)(10,27)(11,32)(12,23)(13,24)(14,25)(16,26)(17,28)(21,31)(22,29);(1,4)(2,10)(3,13)(5,16)(6,22)(7,25)(8,9)(11,30)(12,18)(14,20)(15,19)(17,26)(21,29)(23,31)(24,27)(28,32)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,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=ei,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, 50889);