Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #447 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #8310 |
| prime radical | #3584 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
796 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,7,13,6,12,23,11,9,14,5,22,8,3,15,17)(4,16,18,26,20,10,25,27,21,28,24,31,19,29,32,30);(1,3)(2,6)(4,16)(5,11)(7,17)(8,14)(9,12)(10,20)(13,23)(15,22)(18,26)(19,29)(21,28)(24,31)(25,27)(30,32);(1,4)(2,10)(3,16)(5,18)(6,20)(7,24)(8,19)(9,21)(11,26)(12,28)(13,30)(14,29)(15,25)(17,31)(22,27)(23,32)Polycyclic presentation:
2^9|a^2=f,d^2=gi,e^2=i,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dg,g^b=gi,e^c=ei,f^c=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36721);