Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #478918 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17175 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
808 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,13,7,11,6,12)(3,14,17,10,18,8,16,26)(4,19,24,29,22,30,23,9)(15,27,32,31,21,28,20,25);(1,3,5,16,7,18,6,17)(2,8,13,10,11,14,12,26)(4,20,24,21,22,32,23,15)(9,27,19,25,29,28,30,31);(1,4,5,24,7,22,6,23)(2,9,13,19,11,29,12,30)(3,15,16,20,18,21,17,32)(8,25,10,28,14,31,26,27);(1,3)(2,10)(4,21)(5,17)(6,16)(7,18)(8,13)(9,28)(11,26)(12,14)(15,22)(19,25)(20,24)(23,32)(27,29)(30,31)Polycyclic presentation:
2^9|a^2=eg,b^2=ef,c^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=def,g^b=gi,d^c=de,e^c=ei,f^c=fi,g^c=gi,f^d=fi,h^d=hi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386114);