Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500661 |
| 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×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1344 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,6,4,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,25)(21,26)(22,24)(27,30)(28,32)(29,31);(1,2)(3,6)(4,8)(5,7)(17,19,22,29)(18,23,24,31)(20,28,21,27)(25,32,26,30);(1,3,5,4)(2,6,7,8)(10,13)(14,16)(19,27)(20,21)(23,30)(25,26)(28,29)(31,32);(2,7)(6,8)(10,13)(14,16)(18,24)(23,31)(25,26)(30,32);(1,4,5,3)(2,8,7,6)(9,11,12,15)(10,14,13,16)(17,20,22,21)(18,25,24,26)(19,28,29,27)(23,32,31,30)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=f,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,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, 7420253);