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 | |
4932 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)(19,27)(20,29)(21,31)(22,23)(24,28)(25,30)(26,32);(3,7)(6,8)(10,14)(11,15)(18,23)(21,32)(24,27)(25,29);(1,2,5,4)(3,6,7,8)(9,11,13,15)(10,12,14,16)(18,23)(19,28)(25,29)(26,31);(9,12)(10,15)(11,14)(13,16)(17,19)(18,24)(20,26)(21,29)(22,28)(23,27)(25,32)(30,31);(1,3,5,7)(2,6,4,8)(17,20)(18,25)(19,26)(21,27)(22,30)(23,29)(24,32)(28,31);(1,4,5,2)(3,8,7,6)(17,21,22,32)(18,26,23,31)(19,29,28,25)(20,27,30,24)Polycyclic presentation:
2^9|c^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8663304);