Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #390401 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14571 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1076 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11)(3,12)(4,8)(5,10,19,9)(7,14)(13,26,29,32)(15,21)(16,25)(17,23)(18,24)(20,28,31,27)(22,30);(1,3)(2,7)(4,13)(5,14)(6,16)(8,20)(9,12)(10,25)(11,21)(15,19)(17,27)(18,29)(22,26)(23,32)(24,31)(28,30);(1,4)(2,8)(3,13,16,29)(5,17)(6,18)(7,20,21,31)(9,22)(10,23)(11,24)(12,26,25,32)(14,27,15,28)(19,30);(2,9)(3,14)(5,19)(8,22)(10,11)(12,25)(13,27)(15,16)(17,30)(23,24)(26,32)(28,29)Polycyclic presentation:
2^9|a^2=h,c^2=gh,e^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,e^b=ei,h^b=hi,g^c=gi,h^c=hi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 297597);