Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| automorphism | Aut(G) order 2097152 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1710 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)(3,4)(5,7)(6,8)(10,12)(14,15)(19,24)(21,23)(25,26,28,30)(27,31,32,29);(1,3,6,7)(2,5,8,4)(9,10)(11,15)(12,13)(14,16)(17,18,22,20)(19,21,24,23)(27,32)(29,31);(1,4,6,5)(2,3,8,7)(9,11,13,16)(10,14,12,15)(17,19,22,24)(18,23,20,21)(25,26)(27,29)(28,30)(31,32);(1,5,6,4)(2,7,8,3)(9,12,13,10)(11,15,16,14)(17,20,22,18)(19,21,24,23);(1,2)(3,5)(4,7)(6,8)(9,11)(10,15)(12,14)(13,16)(17,21)(18,19)(20,24)(22,23)(25,27,28,32)(26,29,30,31)Polycyclic presentation:
2^9|a^2=fgh,b^2=hi,c^2=h,d^2=h,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3970192);