Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143009 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2021 |
| commutator, Frattini, cosocle | C2×C4×Q8 | C23 |
| prime radical | #18401 | C2 |
| LCS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
530 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 530 conjugacy classes of subgroups — too many to draw. Only its 58 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 — 3 generators on 32 points:
(1,2)(3,11)(4,7)(5,9)(6,10)(8,13)(12,16)(14,15)(17,18,21,22)(19,25,28,32)(20,29,31,24)(23,30,26,27);(2,7)(3,12)(4,5)(9,10)(11,15)(13,14)(17,19)(18,23)(20,30)(21,28)(22,26)(24,32)(25,29)(27,31);(1,3)(2,8)(4,12)(5,13)(6,14)(7,11)(9,15)(10,16)(18,22)(19,26)(23,28)(24,29)(25,27)(30,32)Polycyclic presentation:
2^9|a^2=i,d^2=g,e^2=gi,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=egi,f^a=fghi,c^b=cf,d^b=dg,e^b=eg,f^b=fg,h^b=hi,d^c=dh,e^c=egi,f^c=fg,f^d=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50206);