Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C2×#2543 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2758 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,9)(8,11)(10,12)(14,15)(16,20)(21,22)(23,28)(24,25)(26,27);(6,7)(10,11)(13,14,17,15)(16,18,20,19)(21,23)(22,25)(24,26)(27,28);(1,3,4,2)(6,7)(10,11)(14,15)(18,19)(22,27)(23,24);(5,7,9,6)(8,11,12,10)(13,15,17,14)(16,19,20,18)(21,24,26,23)(22,25,27,28);(5,8)(6,10)(7,11)(9,12)(13,16,17,20)(14,18,15,19)(21,25)(22,23)(24,27)(26,28)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=i,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=dgh,e^a=ef,c^b=ch,d^b=dg,d^c=df92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 489849);