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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5462 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)(3,4)(5,6)(7,8)(17,18)(19,24)(20,26)(21,27)(22,29)(23,25)(28,32)(30,31);(2,6)(3,8)(9,10)(11,15)(12,13)(14,16)(17,19,23,21)(18,24,25,27)(20,28,30,29)(22,26,32,31);(3,8)(4,7)(10,13)(11,16)(18,25)(19,21)(20,30)(22,32);(1,3,5,8)(2,7,6,4)(9,11,12,16)(10,14,13,15)(17,20,23,30)(18,26,25,31)(19,29,21,28)(22,27,32,24);(1,3)(2,4)(5,8)(6,7)(9,11)(10,14)(12,16)(13,15)(17,21,23,19)(18,27,25,24)(20,29,30,28)(22,31,32,26);(1,4)(2,3)(5,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,22)(18,28)(19,26)(20,27)(21,31)(23,32)(24,30)(25,29)Polycyclic presentation:
2^9|b^2=h,d^2=gi,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8969462);