Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5024635 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
3132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(5,6,7,8)(9,10,13,14)(11,12,15,16)(17,18,22,23)(19,20,26,24)(21,29,32,25)(27,31,30,28);(9,11)(10,12)(13,15)(14,16)(18,23)(19,26)(25,29)(27,30);(6,8)(10,14)(11,15)(17,19,22,26)(18,24,23,20)(21,30,32,27)(25,28,29,31);(1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(13,15)(14,16)(17,20,22,24)(18,19,23,26)(21,31,32,28)(25,30,29,27);(9,12)(10,11)(13,16)(14,15)(17,21,22,32)(18,25,23,29)(19,27,26,30)(20,28,24,31)Polycyclic presentation:
2^9|a^2=g,c^2=f,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^c=dg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4931854);