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 | |
4508 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10,14,12)(11,17,19,15)(13,16,21,20)(18,23,24,22)(25,26)(27,32)(28,34)(29,37)(30,36)(31,33)(35,40)(38,39);(3,7)(6,8)(10,12)(15,17)(16,20)(22,23)(25,27,31,30)(26,32,33,36)(28,37,38,35)(29,39,40,34);(1,2,5,4)(3,6,7,8)(13,21)(16,20)(18,24)(22,23)(26,33)(27,30)(34,39)(35,37);(1,3,5,7)(2,6,4,8)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,28)(26,34)(27,35)(29,36)(30,37)(31,38)(32,40)(33,39);(1,4,5,2)(3,8,7,6)(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22)(25,29)(26,35)(27,34)(28,36)(30,39)(31,40)(32,38)(33,37);(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)(25,30,31,27)(26,36,33,32)(28,37,38,35)(29,39,40,34)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=hi,d^2=ghi,e^2=ghi,f^2=gi,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, 8178330);