Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C2×C4 | C43 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1806 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)(9,10,13,18,14,19,15,17)(11,20,22,24,23,12,21,16);(2,4)(5,6,7,8)(9,11,14,23)(10,16,19,24)(12,18,20,17)(13,22,15,21);(9,12,15,16,14,20,13,24)(10,11,18,21,19,23,17,22)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,d^2=i,e^2=i,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,h^a=hi,d^b=di,e^b=ei,d^c=di,f^c=fi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72238739, #72238748, #72238750, … |
| maximal normal subgroup of | #72238739, #72238748, #72238750, … |
| maximal quotient of | #72239002 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30700);