Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2091 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,7)(3,5,4,8)(10,13)(11,15)(17,18)(19,20)(21,24)(22,23)(25,26,29,32)(27,28,31,30);(2,7)(5,8)(11,15)(14,16)(17,19,21,23)(18,22,24,20)(27,31)(28,30);(1,3,6,4)(2,8,7,5)(9,10)(11,14)(12,13)(15,16)(17,20,21,22)(18,19,24,23)(25,27,29,31)(26,30,32,28);(1,4,6,3)(2,5,7,8)(9,11,12,15)(10,14,13,16)(17,18)(19,20)(21,24)(22,23)(25,28,29,30)(26,31,32,27);(1,5)(2,4)(3,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,20)(18,23)(19,24)(21,22)(25,26)(27,30)(28,31)(29,32)Polycyclic presentation:
2^9|a^2=fh,b^2=ghi,c^2=fgi,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3850623);