Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2735949 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1422 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,7)(5,8)(9,10)(11,13)(12,15)(14,16)(19,23)(22,24)(25,26,29,28)(27,30,31,32);(1,3,4,8)(2,5,7,6)(9,11,14,12)(10,15,16,13)(27,31)(30,32);(3,8)(5,6)(10,16)(11,12)(17,18,21,20)(19,22,23,24)(25,27,29,31)(26,30,28,32);(1,3)(2,6)(4,8)(5,7)(9,12,14,11)(10,15,16,13)(17,19)(18,22)(20,24)(21,23)(25,26,29,28)(27,30,31,32);(1,2)(3,5)(4,7)(6,8)(9,13,14,15)(10,12,16,11)(17,20,21,18)(19,24,23,22)(25,28,29,26)(27,32,31,30)Polycyclic presentation:
2^9|a^2=fh,b^2=f,c^2=fhi,d^2=fgh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2667551);