Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #384421 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14379 |
| 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 | |
947 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)(3,14,17,26)(4,18)(5,24,16,9)(6,11)(7,12)(8,13)(10,22)(15,30,31,25)(19,27,20,32)(21,28)(23,29);(1,3)(2,9)(4,19)(5,7)(6,16)(8,17)(10,27)(11,26)(12,14)(13,24)(15,22)(18,25)(20,23)(21,31)(28,32)(29,30);(1,4)(2,10,13,28)(3,15)(5,20)(6,21)(7,22)(8,23)(9,25,24,30)(11,18,12,29)(14,27,26,32)(16,19)(17,31);(1,5,7,3,8,16,6,17)(2,9,12,26,13,24,11,14)(4,20,22,15,23,19,21,31)(10,25,29,32,28,30,18,27)Polycyclic presentation:
2^9|a^2=h,c^2=ef,d^2=ei,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291618);