Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #371873 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14082 |
| 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 | |
1060 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)(3,13,16,8)(4,17)(5,21,22,10)(6,12)(7,25)(9,19)(11,24)(14,23,27,28)(15,18,31,29)(20,26)(30,32);(1,3,6,16)(2,8)(4,10)(5,9,22,17)(7,15,24,31)(11,18)(12,13)(14,30)(19,21)(20,27)(23,26,28,32)(25,29);(1,4,6,19)(2,9)(3,14)(5,18,22,29)(7,20,24,30)(8,23,13,28)(10,15)(11,26)(12,17)(16,27)(21,31)(25,32);(1,5,7,23,6,22,24,28)(2,10,11,27,12,21,25,14)(3,9,15,26,16,17,31,32)(4,18,20,13,19,29,30,8)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=ef,d^2=g,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,h^b=hi,e^c=ei,f^c=fi,e^d=ei,f^d=fi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 279069);