Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6885065 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #30118 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1466 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)(17,18,21,24,20,23,28,29)(19,25,27,32,26,22,30,31);(1,3,4,8)(2,5,6,7)(10,13)(11,15)(17,19,20,26)(18,22,23,25)(21,27,28,30)(24,31,29,32);(3,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,23)(21,29)(24,28)(25,26)(27,32)(30,31);(2,6)(5,7)(10,13)(14,16)(18,23)(19,26)(24,29)(27,30);(9,11,12,15)(10,14,13,16)(19,26)(22,25)(27,30)(31,32)Polycyclic presentation:
2^9|a^2=fgi,b^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6873737);