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 | |
4966 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)(11,12)(13,16)(14,15)(17,18)(19,24)(20,26)(21,27)(22,28)(23,25)(29,32)(30,31);(1,2,5,6)(3,4,7,8)(10,15)(12,13)(17,19)(18,24)(20,21)(22,25)(23,28)(26,27)(29,30)(31,32);(2,6)(3,7)(9,11,14,16)(10,13,15,12)(18,25)(19,28)(21,29)(26,31);(2,6)(3,7)(9,12,14,13)(10,16,15,11)(17,20)(18,26)(19,29)(21,28)(22,27)(23,30)(24,32)(25,31);(1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,21,23,29)(18,27,25,32)(19,20,28,30)(22,31,24,26);(1,4)(2,3)(5,8)(6,7)(9,13,14,12)(10,11,15,16)(17,22)(18,19)(20,27)(21,31)(23,24)(25,28)(26,29)(30,32)Polycyclic presentation:
2^9|b^2=ghi,c^2=g,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10236613);