Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2503 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)(3,7,8,6)(9,10,13,12)(11,15,16,14)(17,18,21,23)(19,20,24,22)(25,26,29,31)(27,28,32,30);(10,12)(14,15)(17,19,21,24)(18,22,23,20)(25,27,29,32)(26,30,31,28);(11,16)(14,15)(18,23)(19,24)(26,31)(28,30);(1,3,5,8)(2,6,4,7)(9,11)(10,14)(12,15)(13,16)(17,19)(18,22)(20,23)(21,24)(25,28,29,30)(26,27,31,32);(1,4,5,2)(3,7,8,6)(9,12,13,10)(11,15,16,14)(17,20,21,22)(18,19,23,24)(25,27)(26,30)(28,31)(29,32)Polycyclic presentation:
2^9|a^2=i,b^2=g,d^2=fghi,e^2=ghi,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, 1685495);