Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153194 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13348 |
| commutator, Frattini, cosocle | C2×C4⋊Q8 | C23 |
| prime radical | #33569 | C2 |
| LCS | G ▷ C2×C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊Q8 ◁ G | |
| LECS | G ▷ C2×C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
482 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 482 conjugacy classes of subgroups — too many to draw. Only its 50 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)(3,11,16,7)(4,17)(5,22)(6,8)(9,23)(10,24)(12,29,27,32)(13,25,15,26)(14,30,19,31)(18,20)(21,28);(1,2)(3,12)(4,18)(5,10)(6,8)(7,14)(9,21)(11,19)(13,31)(15,30)(16,27)(17,20)(22,24)(23,28)(25,29)(26,32);(1,3)(2,7)(4,19)(5,13)(6,16)(8,11)(9,27)(10,25)(12,18)(14,21)(15,24)(17,30)(20,29)(22,26)(23,32)(28,31)Polycyclic presentation:
2^9|a^2=h,d^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fghi,g^a=gi,c^b=cf,d^b=dh,e^b=ei,f^b=fi,g^b=gh,d^c=dg,f^c=fi,g^c=gi,h^c=hi,f^d=fh,g^f=gi35 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60390);