Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6342371 |
| automorphism | Aut(G) order 21139292160 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C216 |
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 | |
213509 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 16 points:
(2,4)(6,8)(10,12)(14,16);(5,6)(7,8);(9,10)(11,12);(13,14)(15,16);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)Polycyclic presentation:
2^9|b^a=bf,c^a=cg,d^a=dh,e^a=ei152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6249567);