Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
2026 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,6,7,8)(9,10,12,14)(11,15,16,13)(18,19)(20,24)(26,29)(27,32);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,18,21,19)(20,22,24,23)(25,26)(27,31)(28,29)(30,32);(3,7)(6,8)(10,14)(13,15)(17,18,21,19)(20,23,24,22)(25,27,28,32)(26,30,29,31);(1,2)(3,8)(4,5)(6,7)(9,11)(10,15)(12,16)(13,14)(17,19,21,18)(20,22,24,23)(25,26)(27,30)(28,29)(31,32);(1,3,4,7)(2,6,5,8)(9,10)(11,13)(12,14)(15,16)(17,20)(18,22)(19,23)(21,24)(25,27)(26,31)(28,32)(29,30)Polycyclic presentation:
2^9|a^2=gi,b^2=fghi,c^2=fgi,d^2=fghi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3492839);