Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
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 | |
6708 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 36 points:
(2,4)(5,6)(7,11)(8,10)(9,16)(12,17)(13,14)(15,20)(18,19)(25,33)(29,34)(30,35)(32,36);(1,2,3,4)(5,6,10,8)(7,11,13,14)(9,12,17,16)(15,18,19,20)(21,22,26,27)(23,24,28,31)(25,29,33,34)(30,32,35,36);(6,8)(7,13)(12,16)(15,19)(22,27)(24,31)(29,34)(32,36);(5,7,10,13)(6,11,8,14)(9,15,17,19)(12,18,16,20)(21,23,26,28)(22,24,27,31)(25,30,33,35)(29,32,34,36);(5,8,10,6)(7,14,13,11)(9,16,17,12)(15,20,19,18)(21,24)(22,28)(23,27)(25,32)(26,31)(29,35)(30,34)(33,36);(5,9,10,17)(6,12,8,16)(7,15,13,19)(11,18,14,20)(21,25,26,33)(22,29,27,34)(23,30,28,35)(24,32,31,36)Polycyclic presentation:
2^9|b^2=ghi,d^2=i,e^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7619823);