Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469989 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1668 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(9,10,13,15)(11,16,12,14)(18,23)(20,21)(24,30)(25,29)(26,28)(31,32);(11,12)(14,16)(17,18)(19,24)(20,23)(21,29)(22,25)(26,31)(27,32)(28,30);(1,3,4,8)(2,5,6,7)(9,11,13,12)(10,14,15,16)(17,19,22,27)(18,24,25,32)(20,28,21,26)(23,30,29,31);(9,12,13,11)(10,14,15,16)(17,20,22,21)(18,23,25,29)(19,26,27,28)(24,31,32,30);(10,15)(14,16)(19,27)(24,32)(26,28)(30,31)Polycyclic presentation:
2^9|a^2=g,c^2=gi,d^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6391628);