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 | C24 | #120 |
| LCS | G ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C2×#95 ◁ G | |
| LECS | G ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1 | |
495 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 495 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,7)(3,5,8,6)(10,13)(11,15)(14,16)(17,18)(19,23)(20,22)(21,24);(2,5)(3,8)(6,7)(17,19)(18,22,24,23)(20,21);(9,10,12,14)(11,16,15,13)(18,22)(19,20)(23,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=hi,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=eh,e^d=ei44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33311);