Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501126 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
789 conjugacy classes of subgroups; only the 148 normal subgroups are shown.
This group has 789 conjugacy classes of subgroups — too many to draw. Only its 148 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 — 4 generators on 24 points:
(1,2,5,8)(3,7,4,6)(9,10,13,15)(11,18,17,14)(12,20,22,19)(16,21,24,23);(1,3,5,4)(2,6,8,7)(9,11,13,17)(10,14,15,18)(12,21,22,23)(16,19,24,20);(9,12,14,23,13,22,18,21)(10,16,17,20,15,24,11,19);(1,4,5,3)(2,7,8,6)(9,10,13,15)(11,14,17,18)(12,16,22,24)(19,23,20,21)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=f,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fi,g^a=ghi,c^b=cg,f^b=fh,g^b=gi,g^c=ghi,h^c=hi,g^f=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72268162 |
| maximal normal subgroup of | #72268162 |
| maximal quotient of | #72268167 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410561);