Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432838 |
| 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 | #34624 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1680 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)(11,15)(14,16)(18,23)(19,27)(20,22)(24,31)(28,29)(30,32);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,18,21,23)(19,24,28,30)(20,25,22,26)(27,32,29,31);(10,13)(14,16)(17,19,21,28)(18,24,23,30)(20,27,22,29)(25,32,26,31);(9,11,12,15)(10,14,13,16)(18,23)(24,30)(25,26)(31,32);(1,3,4,7)(2,6,5,8)(17,20,21,22)(18,25,23,26)(19,27,28,29)(24,32,30,31)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=h,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,g^a=gi,c^b=ch,d^b=di,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6342508);