Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437412 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1966 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)(3,8)(4,6)(5,7)(11,15)(13,16)(17,18,21,23)(19,25,27,32)(20,28,30,24)(22,29,31,26);(9,10)(11,16)(12,14)(13,15)(17,19,21,27)(18,22,23,31)(20,29,30,26)(24,32,28,25);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16);(10,14)(13,16);(1,4,5,3)(2,6,7,8)(18,23)(22,31)(24,28)(25,32)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=g,e^2=gh,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,d^b=dh,f^b=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6376178);