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 | |
4704 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10)(11,18)(12,16)(13,17)(14,15)(19,20)(21,23)(22,24)(25,26,30,34)(27,31,36,40)(28,37,35,32)(29,33,38,39);(2,6)(3,7)(10,15)(11,17)(16,23)(19,24)(25,27)(26,31)(28,38)(29,35)(30,36)(32,33)(34,40)(37,39);(1,3)(2,4)(5,7)(6,8)(25,28,30,35)(26,32,34,37)(27,29,36,38)(31,39,40,33);(1,4)(2,3)(5,8)(6,7)(9,11)(10,13)(12,19)(14,17)(15,18)(16,22)(20,23)(21,24)(25,27)(26,31)(28,29)(30,36)(32,39)(33,37)(34,40)(35,38);(9,12)(10,16)(11,19)(13,22)(14,21)(15,23)(17,24)(18,20)(25,26)(27,31)(28,32)(29,39)(30,34)(33,38)(35,37)(36,40);(9,13)(10,17)(11,15)(12,20)(14,18)(16,19)(21,22)(23,24)(25,29)(26,33)(27,35)(28,36)(30,38)(31,32)(34,39)(37,40)Polycyclic presentation:
2^9|a^2=i,c^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8598178);