Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #391157 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14617 |
| 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 | |
853 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)(3,8,15,23)(4,9)(5,10,16,28)(6,11)(7,20)(12,13)(14,30,31,22)(17,24,32,25)(18,29)(19,26)(21,27);(1,3,13,16,6,15,7,5)(2,8,20,10,11,23,12,28)(4,17,18,14,19,32,21,31)(9,24,27,22,26,25,29,30);(1,4)(2,9,11,26)(3,14)(5,17)(6,19)(7,21)(8,22,23,30)(10,25,28,24)(12,29,20,27)(13,18)(15,31)(16,32);(1,5,13,15,6,16,7,3)(2,10,12,23,11,28,20,8)(4,17,18,31,19,32,21,14)(9,25,29,30,26,24,27,22)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=eh,d^2=efg,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=gi,e^d=ei,f^d=fi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 298353);