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 | |
6864 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 36 points:
(2,4)(5,6,10,14)(7,9,13,11)(8,16,18,12)(15,19,20,17);(1,2)(3,4)(22,26)(27,33)(28,34)(35,36);(5,7)(6,11)(8,15)(9,14)(10,13)(12,19)(16,17)(18,20)(21,22,25,26)(23,27,30,33)(24,28,32,34)(29,35,31,36);(5,8,10,18)(6,12,14,16)(7,15,13,20)(9,17,11,19);(5,7)(6,9)(8,15)(10,13)(11,14)(12,17)(16,19)(18,20)(21,23,25,30)(22,27,26,33)(24,31,32,29)(28,36,34,35);(5,9)(6,13)(7,14)(8,17)(10,11)(12,20)(15,16)(18,19)(21,24,25,32)(22,28,26,34)(23,29,30,31)(27,35,33,36)Polycyclic presentation:
2^9|a^2=h,c^2=i,d^2=h,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7561999);