Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432861 |
| 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×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1555 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(11,15)(14,16)(17,18,22,30,21,26,20,25)(19,24,27,23,29,31,28,32);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,13)(15,16);(10,13)(14,16)(17,19)(18,23)(20,27)(21,29)(22,28)(24,30)(25,31)(26,32);(9,11,12,15)(10,14,13,16)(17,19)(18,24)(20,28)(21,29)(22,27)(23,30)(25,32)(26,31);(17,20,21,22)(18,25,26,30)(19,27,29,28)(23,31,32,24)Polycyclic presentation:
2^9|a^2=fgi,b^2=f,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,c^b=ch,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6342824);