Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×C4×C22⋊C4 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
952 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,9,4,7)(3,8,12,16,13,15,11,14)(17,18)(19,26)(20,24)(21,30)(22,25)(23,28)(27,32)(29,31);(2,7,9,10)(8,14,15,16)(17,19,21,29,22,28,20,27)(18,23,24,31,25,26,30,32);(1,3,5,12,6,13,4,11)(2,8,10,16,9,15,7,14);(2,9)(7,10)(8,15)(14,16)(17,20,22,21)(18,24,25,30)(19,27,28,29)(23,31,26,32)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=i,e^2=h,f^2=h,g^2=i,b^a=be,d^a=dh,e^a=eh,g^a=ghi,e^b=ei128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 90083);