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 |
| prime radical | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4523 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,8)(4,6)(5,7)(9,10,12,14)(11,15,16,13)(19,27)(21,29)(24,28)(25,30);(1,3,5,6)(2,4,7,8)(9,11,12,16)(10,13,14,15)(17,18,23,20)(19,28,27,24)(21,25,29,30)(22,26,32,31);(1,2)(3,8)(4,6)(5,7)(17,19,23,27)(18,24,20,28)(21,22,29,32)(25,31,30,26);(17,20,23,18)(19,24,27,28)(21,30,29,25)(22,31,32,26);(1,2)(3,4)(5,7)(6,8)(17,21)(18,25)(19,22)(20,30)(23,29)(24,31)(26,28)(27,32);(1,4)(2,6)(3,7)(5,8)(17,22)(18,26)(19,29)(20,31)(21,27)(23,32)(24,25)(28,30)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=h,d^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7759932);