Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
1999 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,4,8,7)(9,10,13,12)(11,14,16,15)(17,18)(19,20)(21,22)(23,24)(27,31)(29,32);(1,3)(2,4)(5,8)(6,7)(9,11,13,16)(10,14,12,15)(18,22)(20,23)(25,26,28,30)(27,29,31,32);(2,6)(3,8)(10,12)(14,15)(17,19)(18,23)(20,22)(21,24)(25,27,28,31)(26,29,30,32);(1,3)(2,7)(4,6)(5,8)(9,12,13,10)(11,15,16,14)(17,19)(18,20)(21,24)(22,23)(25,27)(26,29)(28,31)(30,32);(1,4,5,7)(2,3,6,8)(9,11)(10,14)(12,15)(13,16)(17,20)(18,24)(19,22)(21,23)(25,26)(27,29)(28,30)(31,32)Polycyclic presentation:
2^9|a^2=fghi,b^2=fgh,c^2=hi,d^2=fgi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5788110);