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 | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1868 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)(3,4)(5,7)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,24)(19,23,28,31)(20,29,22,25)(26,30,32,27);(1,3)(2,6)(4,7)(5,8)(9,11,12,15)(10,13,14,16)(17,18)(19,26)(20,25)(21,24)(22,29)(23,27)(28,32)(30,31);(1,4,5,6)(2,3,7,8)(10,14)(13,16)(17,19,21,28)(18,23,24,31)(20,27,22,30)(25,32,29,26);(2,7)(3,8)(10,14)(13,16)(18,24)(23,31)(25,29)(26,32);(17,20,21,22)(18,25,24,29)(19,27,28,30)(23,32,31,26)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^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, 7411643);