Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #93860 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7092 |
| commutator | C82 | C2×C4 |
| Frattini, cosocle | #1311 | C22 |
| prime radical | #1712 | C4 |
| LCS | G ▷ C82 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #1311 ◁ G | |
| LECS | G ▷ #1311 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,3,9,4,16,14,28,5,20,15,32,18,7,22,23,6,11,13,29,19,8,25,26,21,10,30,27,17,24,12,31);(2,7)(3,12)(4,17)(5,21)(8,10)(9,27)(11,24)(13,22)(14,30)(15,25)(16,20)(18,19)(26,28)(29,32)Polycyclic presentation:
2^9|a^2=d,c^2=fh,d^2=ef,e^2=g,f^2=h,g^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fgi,g^a=gi,h^a=hi,c^b=cf,d^b=defgi,e^b=egi,f^b=fh,g^b=gi,d^c=di47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1056);