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 | |
4813 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)(12,16)(14,15)(18,23)(19,26)(25,31)(27,28);(1,2)(3,8)(4,6)(5,7)(10,15)(11,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,32)(28,29)(30,31);(1,3)(2,4)(5,6)(7,8)(9,11,14,16)(10,12,15,13)(19,26)(21,24)(27,28)(29,32);(3,6)(4,8)(9,12,14,13)(10,11,15,16)(17,19,22,26)(18,24,23,21)(20,28,30,27)(25,29,31,32);(1,4,5,8)(2,6,7,3)(9,10)(11,12)(13,16)(14,15)(17,20,22,30)(18,25,23,31)(19,27,26,28)(21,29,24,32);(1,2)(3,8)(4,6)(5,7)(9,13,14,12)(10,16,15,11)(17,21,22,24)(18,26,23,19)(20,29,30,32)(25,28,31,27)Polycyclic presentation:
2^9|c^2=h,d^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8819779);