Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #459 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1982 |
| prime radical | C2×#493 | C4 |
| 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 | |
500 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 500 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) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5,7,8)(9,10,15,19,14,20,12,23,17,21,13,18,16,11,22,24);(1,2,6,5,4,7,3,8)(9,11,17,20)(10,14,21,16)(12,23,22,24)(13,19,15,18);(9,12,16,15,17,22,14,13)(10,18,20,24,21,19,11,23)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=ei,d^2=gi,e^2=i,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=ei,d^b=dg,f^b=fg,g^b=gi,h^b=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72216353, #72216421, #72216424, … |
| maximal normal subgroup of | #72216353, #72216421, #72216424, … |
| maximal quotient of | #72216299 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48045);