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 | #39375 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4728 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,6,7)(3,5,8,4)(9,10)(11,15)(12,13)(14,16)(17,18)(19,24)(20,29)(21,30)(22,23)(25,27)(26,31)(28,32);(2,7)(4,5)(10,13)(14,15)(18,23)(24,32)(25,29)(26,30);(1,3)(2,4)(5,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,28)(18,24,23,32)(20,21,27,31)(25,26,29,30);(1,4,6,5)(2,8,7,3)(9,11,12,16)(10,14,13,15)(17,20,22,27)(18,25,23,29)(19,21,28,31)(24,26,32,30);(1,5,6,4)(2,3,7,8)(17,18)(19,24)(20,25)(21,26)(22,23)(27,29)(28,32)(30,31);(1,3)(2,4)(5,7)(6,8)(17,21)(18,26)(19,27)(20,28)(22,31)(23,30)(24,29)(25,32)Polycyclic presentation:
2^9|a^2=h,c^2=ghi,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8015909);