Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1566 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16);(2,5)(6,8)(17,18,22,26)(19,27,20,29)(21,31,32,25)(23,30,24,28);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,19)(18,23)(20,22)(21,30)(24,26)(25,29)(27,31)(28,32);(17,20,22,19)(18,24,26,23)(21,28,32,30)(25,27,31,29);(9,11,12,15)(10,14,13,16)(19,20)(23,24)(27,29)(28,30)Polycyclic presentation:
2^9|b^2=i,c^2=f,d^2=i,e^2=g,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,d^c=di,e^c=ei,h^c=hi,e^d=ei,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6362687);