Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #59033 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4854 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(10,14)(15,16)(17,18)(19,20)(21,29)(22,26)(23,24)(25,31)(27,32)(28,30);(1,2,6,4)(3,7,8,5)(9,10)(11,15)(12,16)(13,14)(17,19,23,22)(18,20,24,26)(21,27,31,30)(25,28,29,32);(2,4)(3,8)(18,24)(20,26)(25,29)(28,32);(1,3,6,8)(2,7,4,5)(17,20,23,26)(18,19,24,22)(21,28,31,32)(25,27,29,30);(1,4,6,2)(3,5,8,7)(9,11,13,12)(10,15,14,16)(17,21,23,31)(18,25,24,29)(19,27,22,30)(20,28,26,32);(1,5)(2,3)(4,8)(6,7)(9,12,13,11)(10,16,14,15)(17,22,23,19)(18,26,24,20)(21,30,31,27)(25,32,29,28)Polycyclic presentation:
2^9|b^2=h,d^2=h,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7618333);