Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1901 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,6,7,8)(9,10)(11,16)(12,13)(14,15)(18,21)(19,23)(27,28)(30,31);(2,4)(6,8)(9,11)(10,13)(12,14)(15,16)(19,23)(22,24)(25,26,29,32)(27,30,28,31);(3,7)(6,8)(10,15)(11,12)(17,18)(19,22)(20,21)(23,24)(25,27,29,28)(26,30,32,31);(1,3,5,7)(2,6,4,8)(9,12,14,11)(10,16,15,13)(17,19)(18,22)(20,23)(21,24)(25,26)(27,30)(28,31)(29,32);(1,4,5,2)(3,8,7,6)(9,13,14,16)(10,11,15,12)(17,18)(19,22)(20,21)(23,24)(25,28,29,27)(26,31,32,30)Polycyclic presentation:
2^9|a^2=fh,b^2=gi,c^2=gi,d^2=f,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1607349);