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) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2052 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,15)(12,13)(14,16)(18,22)(19,24)(25,26,28,30)(27,31,32,29);(10,12)(14,15)(17,18)(19,20)(21,22)(23,24)(25,26,28,30)(27,29,32,31);(11,16)(14,15)(17,19,21,24)(18,23,22,20)(25,27)(26,29)(28,32)(30,31);(1,2)(3,5)(4,6)(7,8)(9,11,13,16)(10,14,12,15)(17,19)(18,23)(20,22)(21,24)(25,26)(27,29)(28,30)(31,32);(1,3,4,7)(2,5,6,8)(9,12,13,10)(11,14,16,15)(17,20,21,23)(18,24,22,19)(25,27)(26,29)(28,32)(30,31)Polycyclic presentation:
2^9|a^2=fi,b^2=gh,c^2=h,d^2=fh,e^2=ghi,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, 3140854);