Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129437 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1782 |
| commutator | C23×C4 | C22×C4 |
| Frattini, prime radical, cosocle | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C2×#518 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
977 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 977 conjugacy classes of subgroups — too many to draw. Only its 142 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:
(9,10)(11,18)(12,16,22,14)(13,17)(15,20)(19,24,21,23);(1,2,4,6)(3,7,8,5)(10,14,17,16)(15,24,18,23);(1,3,4,8)(2,5,6,7)(9,11)(10,15,17,18)(12,21)(13,20)(14,23,16,24)(19,22)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,d^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,g^b=ghi,d^c=di,e^c=ei,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72232947, #72233100, #72233187, … |
| maximal normal subgroup of | #72232947, #72233100, #72233187, … |
| maximal quotient of | #72232929, #72233097, #72236941 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56640);