Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | #17086 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2429 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,10)(7,9,11,12)(13,14)(15,18)(16,17)(19,20)(22,23)(24,28);(2,4)(13,14,16,17)(15,19,20,18)(22,23)(26,27);(1,2)(3,4)(14,17)(18,19)(21,22,25,23)(24,26,28,27);(5,6)(7,9)(8,10)(11,12)(13,14)(15,19)(16,17)(18,20)(21,23,25,22)(24,27,28,26);(5,7,8,11)(6,9,10,12)(13,15)(14,18)(16,20)(17,19)(21,24,25,28)(22,26,23,27)Polycyclic presentation:
2^9|a^2=fgi,b^2=f,c^2=fg,d^2=fg,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3125528);