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 | C26 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3042 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(6,9)(10,12)(13,14,17,19)(15,16,20,18)(21,22,26,28)(23,24,25,27);(13,15,17,20)(14,18,19,16)(21,23,26,25)(22,27,28,24);(2,3)(14,19)(15,20);(1,3,4,2)(5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,20)(18,19)(21,24,26,27)(22,23,28,25);(5,7,8,11)(6,10,9,12)(13,16)(14,15)(17,18)(19,20)(21,25,26,23)(22,27,28,24)Polycyclic presentation:
2^9|a^2=fgh,b^2=f,d^2=fg,e^2=i,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, 4838656);