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) = C27 |
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 | |
2338 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,8,9)(7,10,11,12)(13,14)(15,19)(16,17)(18,20)(21,22,26,23)(24,28,25,27);(6,9)(10,12)(14,16)(15,20)(21,23,26,22)(24,27,25,28);(7,11)(10,12)(14,16)(18,19)(21,24,26,25)(22,27,23,28);(1,2)(3,4)(5,6)(7,12)(8,9)(10,11)(13,15)(14,18)(16,19)(17,20);(5,7)(6,10)(8,11)(9,12)(13,16,17,14)(15,18,20,19)(21,25,26,24)(22,28,23,27)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=h,e^2=g,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, 3946080);