Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C23 | C43 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | #386 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1926 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(9,10)(11,17,18,14)(12,16)(13,15)(19,23,24,21)(20,22);(9,11,12,19,13,18,20,24)(10,14,16,21,15,17,22,23);(2,4)(5,6,7,8)(9,10)(11,14)(12,16)(13,15)(17,18)(19,21)(20,22)(23,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,g^2=i,b^a=bd,c^a=ce,d^a=di,e^b=ei,f^b=fi,d^c=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72226702, #72226728, #72238747, … |
| maximal normal subgroup of | #72226702, #72226728, #72238747, … |
| maximal quotient of | #72226695, #72239105, #72239222, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30661);