Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6439029 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1983 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,8)(3,7,4,6)(9,10,13,17)(11,16,21,15)(12,20,22,24)(14,19,23,18);(1,3,5,4)(2,6,8,7)(9,11,13,21)(10,15,17,16)(12,14,22,23)(18,24,19,20);(1,3,5,4)(2,7,8,6)(9,12,23,21,13,22,14,11)(10,16,19,24,17,15,18,20);(1,4,5,3)(2,6,8,7)(9,10,13,17)(11,20,21,24)(12,16,22,15)(14,18,23,19);(11,21)(12,22)(15,16)(20,24)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=fg,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351929);