Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433049 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(11,15)(14,16)(17,18)(19,24)(20,30)(21,27)(22,28)(23,26)(25,32)(29,31);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,27)(21,30)(22,29)(23,25)(26,32)(28,31);(10,13)(14,16)(17,20,22,32,23,31,21,24)(18,25,27,30,26,19,28,29);(9,11,12,15)(10,14,13,16)(18,26)(20,31)(24,32)(27,28);(17,21,23,22)(18,27,26,28)(19,29,25,30)(20,24,31,32)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=g,d^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,e^b=ei,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6343196);