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 | |
4819 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,6)(7,8)(10,12)(11,15)(17,18)(19,26)(20,28)(21,23)(22,25)(24,30)(27,31)(29,32);(1,2)(3,7)(4,8)(5,6)(9,10,13,12)(11,16,15,14);(1,3,5,4)(2,7,6,8)(9,11,13,15)(10,14,12,16)(17,19,22,21)(18,23,25,26)(20,27,30,29)(24,31,28,32);(1,3,5,4)(2,7,6,8)(9,12,13,10)(11,16,15,14)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,28)(26,32);(1,4,5,3)(2,8,6,7)(17,21,22,19)(18,23,25,26)(20,29,30,27)(24,31,28,32);(17,18)(19,23)(20,24)(21,26)(22,25)(27,31)(28,30)(29,32)Polycyclic presentation:
2^9|b^2=i,c^2=ghi,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7562395);