Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #476 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1948 |
| prime radical | C23 | #428 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
250 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 118 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:
(1,2,5,3,4,6,8,7)(10,16,19,17)(11,20)(12,21)(13,14)(18,24,23,22);(1,3,5,6,4,7,8,2)(9,10,14,23,15,19,13,18)(11,17,12,22,21,16,20,24);(9,11,13,20,15,21,14,12)(10,17,18,24,19,16,23,22)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=e,d^2=g,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dg,f^b=fi,g^b=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72216353, #72216390, #72216391, … |
| maximal normal subgroup of | #72216353, #72216390, #72216391, … |
| maximal quotient of | #72216306 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47586);