Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #384436 |
| 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 | |
975 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)(4,16)(5,7,19,10)(8,13)(9,23)(14,29,27,32)(15,21)(17,25)(18,24)(20,28)(22,30,26,31);(2,7)(3,13)(4,17)(5,19)(8,21)(10,11)(12,20)(14,30)(16,24)(18,23)(22,26)(27,31);(1,3,5,15,6,13,19,28)(2,8,7,20,11,12,10,21)(4,14,17,31,18,27,23,30)(9,22,16,32,25,26,24,29);(1,4,5,17,6,18,19,23)(2,9,10,24,11,25,7,16)(3,14,15,31,13,27,28,30)(8,22,21,29,12,26,20,32)Polycyclic presentation:
2^9|a^2=h,c^2=ef,d^2=e,e^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,g^c=gi,h^c=hi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291632);