Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13357 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C22×Q8 | C22×C4 |
| Frattini, cosocle | C2×C4×Q8 | C23 |
| socle | C22 | #1889 |
| prime radical | C2×#454 | C4 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
550 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 550 conjugacy classes of subgroups — too many to draw. Only its 78 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:
(1,2,5,7,4,3,6,8)(9,10,11,15,14,19,12,17)(13,22,21,24,23,18,20,16);(1,3,5,8,4,2,6,7)(10,15)(11,12)(16,18)(17,19)(20,21)(22,24);(2,3)(7,8)(9,11)(10,16)(12,14)(13,21)(15,22)(17,18)(19,24)(20,23)Polycyclic presentation:
2^9|a^2=f,b^2=df,d^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,e^a=egi,g^a=gi,c^b=ch,d^b=di,e^b=egi,f^b=fi,d^c=dg,e^c=ei,f^c=fg,e^d=ei,f^e=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44049);