Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| 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 | |
2205 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,15)(12,16)(13,14)(17,18,21,24)(19,22,23,20)(25,26,28,31)(27,30,32,29);(1,3)(2,6)(4,5)(7,8)(9,10,13,14)(11,15,16,12)(17,19,21,23)(18,20,24,22)(25,27,28,32)(26,29,31,30);(3,4)(6,7)(10,14)(12,15)(17,19)(18,22)(20,24)(21,23)(25,27,28,32)(26,29,31,30);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,14)(13,16)(17,19)(18,22)(20,24)(21,23)(25,26)(27,29)(28,31)(30,32);(1,4,5,3)(2,7,8,6)(9,12,13,15)(10,16,14,11)(17,20,21,22)(18,23,24,19)(25,27)(26,30)(28,32)(29,31)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=fh,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3805325);