Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #481947 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#1514 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C2×M16 | C24 |
| prime radical | #60710 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2873 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
718 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)(3,13)(4,17,21,7)(5,10)(6,12)(8,16)(9,18,23,32)(11,20)(14,26)(15,25,19,27)(22,31,28,30)(24,29);(2,7)(4,18)(5,20)(8,22)(9,11)(10,23)(12,17)(13,27)(15,30)(16,29)(19,31)(21,32)(24,28)(25,26);(1,3,6,14)(2,8)(4,19,21,15)(5,16,20,29)(7,22)(9,27)(10,26)(11,13)(12,24)(17,28)(18,31,32,30)(23,25);(3,14)(8,24)(13,26)(15,19)(16,29)(22,28)(25,27)(30,31)Polycyclic presentation:
2^9|a^2=f,c^2=gh,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=ch,e^a=eg,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,h^b=hi,d^c=di,e^c=ei,f^e=fi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 389143);