Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144083 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2011 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #33779 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
413 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 413 conjugacy classes of subgroups — too many to draw. Only its 70 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)(3,9)(4,12)(5,8)(6,10)(7,14)(11,16)(13,15)(18,23)(19,26)(20,30)(22,29)(24,32)(25,28)(27,31);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(17,18,21,24)(19,27,20,28)(22,31,29,25)(23,30,32,26);(4,14)(6,15)(7,12)(10,13)(17,18)(19,28)(20,27)(21,24)(22,31)(23,30)(25,29)(26,32)Polycyclic presentation:
2^9|b^2=gh,d^2=gi,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fghi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=ehi,g^b=gi,h^b=hi,d^c=dgi,e^c=eh,g^c=gi,h^c=hi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51301);