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 | |
6626 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 36 points:
(1,2,3,4)(5,6,11,13)(7,15,9,12)(8,14,18,10)(16,17,19,20)(23,24)(25,30)(31,32)(33,36);(1,2)(3,4)(5,7,11,9)(6,12,13,15)(8,16,18,19)(10,17,14,20)(21,22,27,29)(23,30,24,25)(26,34,35,28)(31,33,32,36);(6,13)(8,18)(12,15)(16,19)(21,23,27,24)(22,25,29,30)(26,31,35,32)(28,33,34,36);(5,8)(6,14)(7,16)(9,19)(10,13)(11,18)(12,20)(15,17)(21,24,27,23)(22,25,29,30)(26,32,35,31)(28,33,34,36);(5,9,11,7)(6,15,13,12)(8,19,18,16)(10,20,14,17)(21,25)(22,23)(24,29)(26,33)(27,30)(28,31)(32,34)(35,36);(5,10,11,14)(6,8,13,18)(7,17,9,20)(12,16,15,19)(21,26,27,35)(22,28,29,34)(23,31,24,32)(25,33,30,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=i,d^2=i,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7650652);