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) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3222 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(3,7)(10,11)(12,16)(17,18,21,19)(20,22,24,23)(26,29)(27,31);(1,2)(3,8)(4,7)(5,6)(10,11)(14,15)(18,19)(22,23)(27,31)(30,32);(2,6)(4,8)(9,10,13,11)(12,14,16,15)(17,18,21,19)(20,23,24,22)(25,26)(27,30)(28,29)(31,32);(1,3)(2,4)(5,7)(6,8)(9,11,13,10)(12,15,16,14)(17,19,21,18)(20,22,24,23)(25,27)(26,30)(28,31)(29,32);(1,4)(2,7)(3,6)(5,8)(9,12)(10,14)(11,15)(13,16)(17,20)(18,22)(19,23)(21,24)(25,26)(27,30)(28,29)(31,32)Polycyclic presentation:
2^9|a^2=fgi,c^2=ghi,d^2=ghi,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, 3460234);