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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2106 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)(4,7)(5,6)(9,10,12,14)(11,15,16,13)(17,18,21,22)(19,23,24,20)(25,26,29,32)(27,28,31,30);(2,6)(7,8)(11,16)(13,15)(17,19,21,24)(18,20,22,23)(27,31)(28,30);(1,3,5,4)(2,7,6,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,22)(19,23,24,20)(25,27,29,31)(26,30,32,28);(1,4,5,3)(2,8,6,7)(9,11)(10,13)(12,16)(14,15)(17,20)(18,19)(21,23)(22,24)(25,28,29,30)(26,31,32,27);(1,2)(3,8)(4,7)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,31)(29,32)Polycyclic presentation:
2^9|a^2=fg,b^2=ghi,c^2=fgi,d^2=fh,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, 3856117);