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 | |
6646 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)(7,12)(8,11)(9,13)(10,15)(14,18)(16,19)(17,20)(22,27)(23,31)(29,30)(32,33);(1,2)(3,4)(22,27)(28,34)(29,30)(35,36);(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(21,22,26,27)(23,28,31,34)(24,29,25,30)(32,35,33,36);(5,8,9,16)(6,11,13,19)(7,14,15,20)(10,17,12,18)(21,23,26,31)(22,28,27,34)(24,32,25,33)(29,35,30,36);(5,7)(6,12)(8,14)(9,15)(10,13)(11,18)(16,20)(17,19)(21,24,26,25)(22,29,27,30)(23,32,31,33)(28,35,34,36);(5,6)(7,10)(8,11)(9,13)(12,15)(14,17)(16,19)(18,20)(21,25,26,24)(22,30,27,29)(23,33,31,32)(28,36,34,35)Polycyclic presentation:
2^9|c^2=i,d^2=h,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7564145);