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 | |
5047 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,12)(13,14)(15,16)(17,18,23,25)(19,27,28,24)(20,30,29,26)(21,31,32,22);(1,2)(3,5)(4,6)(7,8)(17,19)(18,24)(20,22)(21,26)(23,28)(25,27)(29,31)(30,32);(10,14)(11,15)(18,25)(19,28)(20,29)(21,32);(9,11,13,15)(10,12,14,16)(17,20,23,29)(18,26,25,30)(19,22,28,31)(21,27,32,24);(9,12)(10,15)(11,14)(13,16)(17,21,23,32)(18,22,25,31)(19,26,28,30)(20,27,29,24);(1,3,4,7)(2,5,6,8)(17,22)(18,21)(19,29)(20,28)(23,31)(24,30)(25,32)(26,27)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7895182);