Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667484 |
| 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 | #2428 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#126 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1417 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)(19,26)(20,23)(21,24)(22,25)(27,32)(28,30)(29,31);(10,13)(11,16)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32);(2,5)(6,8)(9,11,12,16)(10,14,13,15)(17,19)(18,23)(20,21)(22,28)(24,26)(25,32)(27,29)(30,31);(1,3,4,7)(2,6,5,8)(10,13)(14,15);(17,20,21,19)(18,23,24,26)(22,27,29,28)(25,32,31,30)Polycyclic presentation:
2^9|c^2=f,d^2=fgi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,h^c=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6608163);