Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #136906 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1879 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C22×D8 | 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 | |
428 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 428 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) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,6,5,8,3,7)(9,10,11,17,13,20,12,16,15,22,24,18,14,21,23,19);(2,6)(3,4)(7,8)(10,16,22,19)(11,12)(13,14)(17,21,18,20)(23,24);(10,17)(11,23)(12,24)(13,14)(16,21)(18,22)(19,20)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,d^2=g,e^2=h,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,d^b=dg,e^b=eh,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=ehi,f^c=fh,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72265050, #72265057, #72265083, … |
| maximal normal subgroup of | #72265050, #72265057, #72265083, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44153);