Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #136371 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1864 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#135 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1770 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
342 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 98 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)(3,5)(4,7)(6,8)(10,11,18,21)(12,13)(14,15)(17,19,22,20)(23,24);(2,5)(3,8)(6,7)(9,10,15,19,16,18,14,20)(11,12,17,23,21,24,22,13);(9,11)(10,12)(13,20)(14,22)(15,17)(16,21)(18,24)(19,23)Polycyclic presentation:
2^9|a^2=f,b^2=fh,d^2=g,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,d^b=dg,e^b=eh,f^b=fi,g^b=gi,h^b=hi,d^c=dh,e^c=ehi,f^c=fi,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72309412, #72309414, #72309425, … |
| maximal normal subgroup of | #72309412, #72309414, #72309425, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43596);