Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30437 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2668 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14417 |
| prime radical | #60469 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1355 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,8,3,7,11)(4,15,18,13,19,26,17,9)(5,10,22,25,23,14,21,20)(16,29,30,24,31,28,32,27);(1,3)(2,8)(4,13)(5,20)(6,11)(7,12)(9,19)(10,22)(14,21)(15,17)(16,28)(18,26)(23,25)(24,32)(27,30)(29,31);(1,4,7,17,8,19,6,18)(2,9,12,15,3,13,11,26)(5,16,21,32,23,31,22,30)(10,24,25,28,14,27,20,29);(1,5)(2,10)(3,14)(4,16)(6,22)(7,21)(8,23)(9,24)(11,20)(12,25)(13,27)(15,28)(17,32)(18,30)(19,31)(26,29)Polycyclic presentation:
2^9|a^2=h,c^2=eg,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,g^c=gi,e^d=ei,f^d=fi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293122);