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 | |
5048 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10)(11,13)(12,14)(15,16)(18,22)(23,25)(24,28)(30,31);(1,2,4,5)(3,6,7,8)(9,11)(10,13)(12,16)(14,15)(17,18)(19,25)(20,24)(21,22)(23,26)(27,31)(28,29)(30,32);(2,5)(6,8)(10,12)(11,15)(19,26)(23,25)(27,32)(30,31);(3,7)(6,8)(11,15)(13,16)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,31)(30,32);(1,2)(3,6)(4,5)(7,8)(9,12,14,10)(11,13,15,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,31)(29,32);(1,3)(2,6)(4,7)(5,8)(9,13)(10,11)(12,15)(14,16)(17,20,21,29)(18,24,22,28)(19,27,26,32)(23,30,25,31)Polycyclic presentation:
2^9|b^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8499381);