Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13343 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2) |
| commutator | C22×D4 | C22×C4 |
| Frattini, prime radical, cosocle | C23×D4 | C23 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
982 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,3)(5,6,7,8)(9,10)(11,17)(12,14)(13,16)(15,19)(18,23)(20,22)(21,24);(1,2,4,3)(10,14)(13,20)(15,21)(17,23);(1,3,4,2)(9,11,12,18)(10,15,14,21)(13,19,20,24)(16,17,22,23)Polycyclic presentation:
2^9|a^2=g,b^2=di,c^2=d,b^a=bd,c^a=ce,f^a=fhi,c^b=cf,e^b=eh,h^c=hi,f^e=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72284303, #72284367, #72284422, … |
| maximal normal subgroup of | #72284303, #72284367, #72284422, … |
| maximal quotient of | #72284759, #72284782, #72286216, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56012);