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 | #34988 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5266 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(3,7)(6,8)(9,10,14,15)(11,13,12,16)(17,18,22,26)(19,25,21,23)(20,24,28,31)(27,32,29,30);(1,2)(3,6)(4,5)(7,8)(10,15)(11,12)(17,19,22,21)(18,23,26,25)(20,27,28,29)(24,30,31,32);(2,5)(6,8)(9,11,14,12)(10,16,15,13)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,32)(26,31);(1,2)(3,6)(4,5)(7,8)(9,12,14,11)(10,13,15,16)(17,21,22,19)(18,25,26,23)(20,27,28,29)(24,30,31,32);(9,13)(10,11)(12,15)(14,16);(1,3)(2,6)(4,7)(5,8)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=h,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8013193);