Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13357 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2) |
| commutator | C22×Q8 | C22×C4 |
| Frattini, cosocle | C23×Q8 | C23 |
| prime radical | C24 | #120 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C22 ▷ C2 ▷ C1 | |
790 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 790 conjugacy classes of subgroups — too many to draw. Only its 146 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:
(2,4)(5,6,7,8)(10,14)(11,18)(12,20)(15,22)(16,23)(17,24);(9,10,11,14,13,17,18,24)(12,16,21,22,20,15,19,23);(1,2,3,4)(10,15)(12,20)(14,22)(16,17)(19,21)(23,24)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=e,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fhi,c^b=cf,f^b=fhi,h^b=hi,d^c=dh,f^c=fi,f^d=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72268164, #72268166, #72268167, … |
| maximal normal subgroup of | #72268164, #72268166, #72268167, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56478);