Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | #117 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1767 |
| prime radical | C23 | C4×M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
316 conjugacy classes of subgroups; only the 166 normal subgroups are shown.
This group has 316 conjugacy classes of subgroups — too many to draw. Only its 166 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:
(1,2,4,7,5,3,6,8)(10,14)(11,18)(15,17)(19,22);(9,10,11,16,12,17,19,24,13,14,18,21,20,15,22,23);(1,3,4,8,5,2,6,7)(10,15,14,17)(11,18)(16,24,21,23)(19,22)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=de,e^2=g,f^2=h,h^2=i,b^a=bd,f^a=fi,c^b=ch,d^b=di,f^c=fi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72216225, #72216228, #72216232, … |
| maximal normal subgroup of | #72216225, #72216228, #72216232, … |
| maximal quotient of | #72216252 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36251);