Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1776 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)(3,5)(4,6)(7,8)(9,10,12,14)(11,13,15,16)(17,18)(19,22)(20,24)(21,23)(25,26)(27,31)(28,32)(29,30);(2,3)(6,7)(17,19,21,24)(18,22,23,20)(25,27,30,28)(26,31,29,32);(4,8)(6,7)(17,20)(18,19)(21,22)(23,24)(25,26,30,29)(27,32,28,31);(1,3,5,2)(4,6,8,7)(9,10)(11,13)(12,14)(15,16)(17,20)(18,19)(21,22)(23,24)(25,28,30,27)(26,31,29,32);(1,4)(2,6)(3,7)(5,8)(9,11,12,15)(10,13,14,16)(17,19)(18,20)(21,24)(22,23)(25,29,30,26)(27,31,28,32)Polycyclic presentation:
2^9|a^2=fgi,b^2=g,c^2=h,d^2=fh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4028732);