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 | |
6811 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)(9,12)(13,15)(14,19)(16,17)(18,20)(21,22)(23,26)(24,29)(25,30)(27,28)(31,33)(32,35)(34,36);(1,2)(3,4)(5,6,10,8)(7,13,15,11)(9,12,17,16)(14,20,18,19)(21,23,27,32)(22,26,28,35)(24,25,31,34)(29,30,33,36);(6,8)(9,17)(11,13)(14,18)(22,28)(26,35)(29,33)(30,36);(5,7)(6,11)(8,13)(9,18)(10,15)(12,20)(14,17)(16,19)(21,24)(22,29)(23,25)(26,30)(27,31)(28,33)(32,34)(35,36);(5,8,10,6)(7,13,15,11)(9,16,17,12)(14,20,18,19)(21,25)(22,30)(23,31)(24,32)(26,33)(27,34)(28,36)(29,35);(5,9)(6,12)(7,14)(8,16)(10,17)(11,19)(13,20)(15,18)(21,26,27,35)(22,23,28,32)(24,30,31,36)(25,33,34,29)Polycyclic presentation:
2^9|b^2=h,e^2=i,f^2=hi,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7998934);