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) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2430 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,3,4)(6,8)(9,10)(11,14)(12,16)(13,15)(17,18,21,22)(19,25,26,23)(20,28,29,24)(27,31,30,32);(1,2,3,4)(9,11,13,16)(10,14,15,12)(18,22)(19,26)(20,29)(31,32);(1,2)(3,4)(9,12)(10,11)(13,14)(15,16)(17,19,21,26)(18,23,22,25)(20,30,29,27)(24,32,28,31);(5,6)(7,8)(9,12)(10,11)(13,14)(15,16)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,32)(30,31);(1,2)(3,4)(9,11)(10,12)(13,16)(14,15)(17,20)(18,24)(19,27)(21,29)(22,28)(23,31)(25,32)(26,30)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=f,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, 3941591);