Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2238 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #481 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1926 |
| prime radical | C22×D8 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
285 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 285 conjugacy classes of subgroups — too many to draw. Only its 94 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)(10,14,15,22)(11,17)(12,20)(16,23,21,19)(18,24);(1,3,4,8,5,2,6,7)(11,18)(14,22)(17,24)(19,23);(9,10,12,21,13,15,20,16)(11,19,17,14,18,23,24,22)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=dg,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,e^b=ei,d^c=di,e^c=eg,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72216353, #72216448, #72216456 |
| maximal normal subgroup of | #72216353, #72216448, #72216456 |
| maximal quotient of | #72216338 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47314);