Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #125705 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×#95 | C23 |
| prime radical | #2656 | C22 |
| LCS | G ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C2×#95 ◁ G | |
| LECS | G ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1 | |
539 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 539 conjugacy classes of subgroups — too many to draw. Only its 62 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:
(2,5)(3,7)(6,8)(10,13)(11,15)(14,16)(17,18)(19,27)(20,25)(21,24)(22,26)(23,28)(29,32)(30,31);(1,2,4,6)(3,8,7,5)(11,15)(13,16)(17,19)(18,23,26,31)(20,30)(21,28)(22,29)(24,32,25,27);(9,10,12,14)(11,16,15,13)(17,20,22,21)(18,24)(19,28,29,30)(23,27)(25,26)(31,32)Polycyclic presentation:
2^9|b^2=fghi,c^2=gi,d^2=f,e^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,g^a=gi,c^b=ci,d^b=dfi,e^b=egi,d^c=dg,e^c=ehi,e^d=ei44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33063);