Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| 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 | |
2052 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,9)(8,11)(10,12)(13,14)(15,16)(17,18)(19,20)(21,22,24,26)(23,27,28,25);(2,4)(6,7)(10,11)(13,15,19,18)(14,17,20,16);(1,2)(3,4)(5,6,9,7)(8,10,12,11)(13,16,19,17)(14,15,20,18);(5,7,9,6)(8,10,12,11)(13,17,19,16)(14,15,20,18)(21,23,24,28)(22,25,26,27);(5,8)(6,10)(7,11)(9,12)(13,18,19,15)(14,17,20,16)Polycyclic presentation:
2^9|a^2=fi,b^2=fg,c^2=f,d^2=i,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2561083);