Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483015 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#1530 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | #97 | C24 |
| prime radical | #60725 | C2 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2875 ◁ G | |
| LECS | G ▷ #97 ▷ C2×C4 ▷ C2 ▷ C1 | |
726 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,10,5,7,19,21,6,11,18,27,20,9,16,26)(3,12,14,29,15,23,31,28,13,25,17,32,30,8,22,24);(2,7)(4,16)(5,20)(8,12)(9,11)(10,27)(14,22)(15,30)(17,31)(18,19)(23,25)(24,28);(1,3)(2,8,11,23)(4,17)(5,15)(6,13)(7,12,9,25)(10,28,27,24)(14,18)(16,31)(19,22)(20,30)(21,32,26,29);(3,13)(8,23)(12,25)(14,17)(15,30)(22,31)(24,28)(29,32)Polycyclic presentation:
2^9|a^2=f,c^2=h,e^2=h,f^2=g,g^2=i,b^a=be,c^a=cg,e^a=eg,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,d^c=di,e^c=ei,f^c=fi,f^e=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390211);