Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4829 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(6,8)(11,15)(13,16)(18,22)(23,29)(24,27)(30,31);(9,10,12,14)(11,13,15,16)(17,18)(19,23)(20,24)(21,22)(25,30)(26,29)(27,28)(31,32);(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19,21,26)(18,23,22,29)(20,25,28,32)(24,30,27,31);(1,2)(3,4)(5,6)(7,8)(17,18)(19,23)(20,27)(21,22)(24,28)(25,31)(26,29)(30,32);(9,11)(10,13)(12,15)(14,16)(17,20)(18,24)(19,25)(21,28)(22,27)(23,30)(26,32)(29,31)Polycyclic presentation:
2^9|b^2=h,c^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4325674);