Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| 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 | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1895 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,7)(3,5,8,4)(9,10)(11,13)(12,14)(15,16)(17,18,23,19)(20,24,30,26)(21,27,22,25)(28,29,32,31);(1,3,6,8)(2,4,7,5)(9,11,12,15)(10,13,14,16)(17,19,23,18)(20,28,30,32)(21,27,22,25)(24,29,26,31);(1,4,6,5)(2,3,7,8)(10,14)(13,16)(17,20)(18,24)(19,26)(21,29)(22,31)(23,30)(25,28)(27,32);(1,5,6,4)(2,3,7,8)(10,14)(13,16);(17,21,23,22)(18,25,19,27)(20,29,30,31)(24,28,26,32)Polycyclic presentation:
2^9|a^2=fi,b^2=fgi,c^2=f,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,h^b=hi,h^c=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7412549);