Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9946 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #392 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#392 |
| prime radical | #60734 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1472 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,6)(4,7)(5,8)(9,12)(10,13)(11,15)(14,16)(17,18)(19,28)(20,25)(21,26)(22,27)(23,30)(24,29)(31,32);(1,3,5,10)(2,6,8,13)(4,9,11,16)(7,14,15,12)(17,19,22,31)(18,23,27,24)(20,29,21,30)(25,28,26,32);(1,4)(2,7)(3,9)(5,11)(6,12)(8,15)(10,16)(13,14)(17,19)(18,24)(20,29)(21,30)(22,31)(23,27)(25,32)(26,28);(17,20,22,21)(18,25,27,26)(19,29,31,30)(23,28,24,32)Polycyclic presentation:
2^9|b^2=i,d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,e^c=ei,f^c=fh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 142240);