Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4928752 |
| 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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2573 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)(5,6,9,11)(7,12,8,10)(13,14,17,18)(15,19,16,20)(21,22,25,26)(23,27,28,24);(5,7,9,8)(6,10,11,12)(14,18)(19,20)(22,26)(23,28);(6,11)(10,12)(13,15,17,16)(14,19,18,20)(21,23,25,28)(22,27,26,24);(1,2)(3,4)(5,6)(7,12)(8,10)(9,11)(13,14)(15,19)(16,20)(17,18)(21,22)(23,27)(24,28)(25,26);(5,8,9,7)(6,10,11,12)(13,16,17,15)(14,20,18,19)(21,24,25,27)(22,28,26,23)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,d^c=dg104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4835998);