Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667836 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#126 ◁ 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,13)(14,16)(17,18,22,23)(19,24,29,26)(20,25,21,30)(27,28,31,32);(10,13)(11,16)(17,19,22,29)(18,24,23,26)(20,27,21,31)(25,28,30,32);(2,5)(6,8)(9,11,12,16)(10,14,13,15)(18,23)(19,26)(20,21)(24,29)(27,28)(31,32);(1,3,4,7)(2,6,5,8)(10,13)(14,15)(17,20)(18,25)(19,27)(21,22)(23,30)(24,28)(26,32)(29,31);(17,21,22,20)(18,25,23,30)(19,28,29,32)(24,27,26,31)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=f,d^2=fgi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6608603);