Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| 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 | C2×#127 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1314 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)(3,4)(5,7)(6,8)(9,10,12,13)(11,15,16,14)(17,18,21,25)(19,26,28,23)(20,30,22,24)(27,32,29,31);(1,3)(2,6)(4,7)(5,8)(17,19,21,28)(18,23,25,26)(20,29,22,27)(24,31,30,32);(10,13)(14,15)(19,27)(20,22)(23,31)(24,30)(26,32)(28,29);(2,7)(3,8)(9,11,12,16)(10,14,13,15)(19,28)(23,26)(27,29)(31,32);(1,4,5,6)(2,3,7,8)(17,20,21,22)(18,24,25,30)(19,29,28,27)(23,31,26,32)Polycyclic presentation:
2^9|a^2=gi,b^2=i,d^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,e^a=ei,c^b=ch,d^b=df,h^b=hi,e^c=ei,h^c=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7418270);