Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512651 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2127 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | #124 | C24 |
| prime radical | #60825 | C2 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C2×#41 ◁ #3500 ◁ G | |
| LECS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
832 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,10)(3,8,13,23)(5,9,15,24)(6,11,16,25)(7,17,20,18)(12,22,26,32)(14,21,28,31)(19,29,30,27);(2,7)(4,15)(5,18)(8,21)(9,11)(13,28)(14,27)(16,17)(19,22)(20,24)(23,32)(26,29);(1,3)(2,8)(4,13)(5,14)(6,12)(7,19)(9,21)(10,23)(11,22)(15,28)(16,26)(17,29)(18,27)(20,30)(24,31)(25,32);(3,12)(8,22)(13,26)(14,27)(19,21)(23,32)(28,29)(30,31)Polycyclic presentation:
2^9|a^2=f,e^2=h,g^2=i,b^a=be,e^a=eg,g^a=gh,h^a=hi,c^b=ch,e^b=eh,f^b=fghi,g^b=gi,d^c=di,e^c=ei,f^e=fhi,g^f=gi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419847);