Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486851 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C4⋊C4 | C24 |
| prime radical | #537 | C23 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
723 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)(3,10)(4,7)(5,9)(6,14)(8,12)(11,15)(13,16)(17,18)(19,23)(20,25)(21,24)(22,27)(26,30)(28,31)(29,32);(1,3)(2,8)(4,11)(5,12)(6,15)(7,13)(9,16)(10,14)(17,19,22,20)(18,23,27,25)(21,29,30,28)(24,31,26,32);(1,4,7,2)(3,11,13,8)(5,9,6,14)(10,12,16,15)(18,24)(21,30)(23,31)(25,32)(26,27)(28,29);(1,5,7,6)(2,9,4,14)(3,12,13,15)(8,16,11,10)(17,20,22,19)(18,25,27,23)(21,29,30,28)(24,32,26,31)Polycyclic presentation:
2^9|b^2=i,c^2=g,d^2=gi,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,h^a=hi,e^b=eg,f^b=fhi,d^c=dg,e^c=eh,f^c=fi,f^e=fi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 394684);