Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8603 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | #117 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1749 |
| prime radical | C24 | #97 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
397 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 397 conjugacy classes of subgroups — too many to draw. Only its 138 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:
(1,2,4,3,5,6,8,7)(10,15,17,16)(18,22,24,23);(9,10,12,18,13,15,21,22,14,17,20,24,11,16,19,23);(1,3,4,6,5,7,8,2)(10,16,17,15)(12,20)(18,22,24,23)(19,21)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=i,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=cg,c^b=ch,d^b=di,e^b=ei,f^c=fi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35475);