Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
2364 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)(9,10,12,14)(11,15,16,13)(17,18)(19,24)(20,26)(21,29)(22,23)(25,30)(27,28)(31,32);(6,8)(9,10,12,14)(11,13,16,15)(18,23)(19,26)(21,30)(28,31);(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19,22,26)(18,24,23,20)(21,27,30,32)(25,31,29,28);(1,2)(3,4)(9,10)(11,13)(12,14)(15,16)(17,20,22,24)(18,19,23,26)(21,31,30,28)(25,32,29,27);(9,11,12,16)(10,13,14,15)(17,21,22,30)(18,25,23,29)(19,27,26,32)(20,28,24,31)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=f,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2976869);