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 | |
7242 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)(7,13)(8,16)(9,10)(11,14)(12,17)(15,18)(19,20)(25,33)(29,34)(30,36)(32,35);(1,2)(3,4)(5,6,9,10)(7,11,14,13)(8,12,17,16)(15,18,20,19)(21,22)(23,24)(25,29)(26,27)(28,31)(30,32)(33,34)(35,36);(6,10)(7,14)(12,16)(15,20)(22,27)(24,28)(29,34)(32,35);(5,6)(7,11)(8,12)(9,10)(13,14)(15,18)(16,17)(19,20)(21,23)(22,28)(24,27)(25,30)(26,31)(29,35)(32,34)(33,36);(5,7,9,14)(6,11,10,13)(8,15,17,20)(12,18,16,19)(21,24,26,28)(22,23,27,31)(25,32,33,35)(29,30,34,36);(5,8)(6,12)(7,15)(9,17)(10,16)(11,18)(13,19)(14,20)(21,25)(22,29)(23,30)(24,32)(26,33)(27,34)(28,35)(31,36)Polycyclic presentation:
2^9|b^2=h,e^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7621701);