Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 7 |
| isoclinism | isoclinic class of #94827 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7529 |
| commutator | C82 | C2×C4 |
| Frattini, cosocle | #1709 | C22 |
| prime radical | #12519 | C2 |
| LCS | G ▷ C82 ▷ C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ C4×C8 ◁ #1709 ◁ G | |
| LECS | G ▷ #1709 ▷ C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
115 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,3,9)(4,16,12,28)(5,20,22,30)(6,10,13,29)(7,23,24,19)(8,25,26,18)(11,15,31,21)(14,32,17,27);(2,7)(3,12)(4,17)(5,21)(8,20)(10,16)(11,27)(13,23)(14,22)(15,25)(18,19)(24,26)(28,32)(30,31)Polycyclic presentation:
2^9|a^2=d,c^2=fgi,e^2=gh,f^2=hi,g^2=i,b^a=bc,c^a=ce,e^a=ef,f^a=fg,g^a=gh,h^a=hi,c^b=cfghi,d^b=defi,e^b=eghi,f^b=fhi,g^b=gi,d^c=dfg,e^d=eghi,f^d=fhi,g^d=gi29 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2023);