Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #763332 |
| 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 | |
2102 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)(3,7)(10,13)(15,16)(17,18)(19,23)(20,21)(22,24)(26,29)(27,31);(1,2,5,4)(3,8,7,6)(9,10,14,13)(11,15,12,16)(18,21)(22,23)(25,26,28,29)(27,30,31,32);(2,4)(6,8)(9,11,14,12)(10,15,13,16)(17,19,20,24)(18,22,21,23)(25,27,28,31)(26,30,29,32);(1,3,5,7)(2,6,4,8)(9,12,14,11)(10,15,13,16)(17,18)(19,22)(20,21)(23,24)(25,26)(27,30)(28,29)(31,32);(1,4,5,2)(3,8,7,6)(9,13,14,10)(11,15,12,16)(17,19)(18,22)(20,24)(21,23)(25,27)(26,30)(28,31)(29,32)Polycyclic presentation:
2^9|b^2=fghi,c^2=fh,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df,e^c=ef92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 674079);