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 | |
4920 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,4)(6,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,26)(20,24)(21,29)(22,23)(25,30)(27,28)(31,32);(3,7)(6,8)(10,14)(11,15)(17,19,22,24)(18,20,23,26)(21,27,30,31)(25,28,29,32);(1,2,5,4)(3,6,7,8)(9,11,13,15)(10,12,14,16)(18,23)(20,26)(21,30)(27,31);(9,12)(10,15)(11,14)(13,16)(17,18)(19,20)(21,25)(22,23)(24,26)(27,28)(29,30)(31,32);(1,3,5,7)(2,6,4,8)(17,20)(18,24)(19,23)(21,28)(22,26)(25,31)(27,29)(30,32);(1,4,5,2)(3,8,7,6)(17,21,22,30)(18,25,23,29)(19,27,24,31)(20,28,26,32)Polycyclic presentation:
2^9|b^2=gi,c^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8961590);