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 | |
5161 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)(3,4)(5,6)(7,8)(10,13)(14,16)(17,18)(19,20)(21,25)(22,23)(24,28)(26,27)(29,30)(31,32);(9,10,12,13)(11,14,15,16)(17,18,22,23)(19,20,24,28)(21,29,30,25)(26,32,31,27);(2,6)(3,7)(18,23)(19,24)(21,30)(27,32);(1,3,5,7)(2,4,6,8)(9,11,12,15)(10,14,13,16)(17,19)(18,20)(21,26)(22,24)(23,28)(25,27)(29,32)(30,31);(1,4)(2,7)(3,6)(5,8)(17,20)(18,24)(19,23)(21,27)(22,28)(25,31)(26,29)(30,32);(17,21)(18,25)(19,26)(20,27)(22,30)(23,29)(24,31)(28,32)Polycyclic presentation:
2^9|b^2=g,d^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7719541);