Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #437530 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15602 |
| 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 | |
948 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,22,6,12,8,13)(3,9,5,11,15,25,21,24)(4,16,19,10,18,31,20,27)(14,28,17,23,29,26,32,30);(1,3,7,5,6,15,8,21)(2,9,13,24,12,25,22,11)(4,17,19,29,18,32,20,14)(10,26,16,23,27,28,31,30);(1,4,8,20,6,18,7,19)(2,10,22,31,12,27,13,16)(3,14,5,17,15,29,21,32)(9,23,24,28,25,30,11,26);(1,5,8,15,6,21,7,3)(2,11,22,9,12,24,13,25)(4,17,20,29,18,32,19,14)(10,26,31,23,27,28,16,30)Polycyclic presentation:
2^9|a^2=eg,b^2=f,c^2=ef,d^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,h^c=hi,e^d=ei,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 344726);