Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628847 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2018 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(19,27)(20,21)(24,31)(25,26)(28,29)(30,32);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,30)(29,32);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(18,23)(24,30)(25,26)(31,32);(17,19)(18,24)(20,27)(21,29)(22,28)(23,30)(25,31)(26,32);(17,20,22,21)(18,25,23,26)(19,27,28,29)(24,31,30,32)Polycyclic presentation:
2^9|b^2=fg,c^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,h^a=hi,c^b=ci,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536795);