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 | |
4858 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,6)(7,8)(10,13)(14,15)(17,18)(19,23)(20,25)(21,29)(22,24)(26,30)(27,32)(28,31);(1,2)(3,7)(4,8)(5,6)(17,19,22,20)(18,23,24,25)(21,27,30,28)(26,31,29,32);(9,10)(11,15)(12,13)(14,16)(18,24)(19,20)(26,29)(27,28);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,20,22,19)(18,25,24,23)(21,28,30,27)(26,32,29,31);(1,3,5,4)(2,7,6,8)(17,18)(19,23)(20,25)(21,26)(22,24)(27,31)(28,32)(29,30);(1,4,5,3)(2,8,6,7)(17,21)(18,26)(19,27)(20,28)(22,30)(23,31)(24,29)(25,32)Polycyclic presentation:
2^9|b^2=i,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^c=dh,e^c=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7577919);