Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
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 | |
4821 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,14,15)(11,17,18,16)(12,21,22,13)(19,20,24,23)(25,26)(27,31)(28,35)(29,30)(32,36)(33,39)(34,37)(38,40);(2,3)(6,7)(10,15)(11,18)(12,22)(20,23)(26,30)(31,37)(32,35)(38,39);(1,3,5,2)(4,7,8,6)(10,15)(13,21)(16,17)(20,23)(25,27,29,34)(26,31,30,37)(28,33,36,40)(32,38,35,39);(1,4)(2,6)(3,7)(5,8)(9,11,14,18)(10,16,15,17)(12,19,22,24)(13,20,21,23)(25,28)(26,32)(27,33)(29,36)(30,35)(31,38)(34,40)(37,39);(9,12)(10,13)(11,19)(14,22)(15,21)(16,20)(17,23)(18,24)(25,26)(27,31)(28,32)(29,30)(33,38)(34,37)(35,36)(39,40);(9,13)(10,12)(11,20)(14,21)(15,22)(16,19)(17,24)(18,23)(25,27)(26,31)(28,33)(29,34)(30,37)(32,38)(35,39)(36,40)Polycyclic presentation:
2^9|a^2=hi,c^2=gh,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8109557);