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 | |
4492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,7)(9,10)(11,12)(13,17)(14,15)(16,19)(18,20)(21,22)(23,24)(25,26)(27,34)(28,38)(29,33)(30,31)(32,35)(36,37)(39,40);(1,2,5,4)(3,7,8,6)(10,15)(12,16)(17,22)(20,23)(26,31)(27,35)(28,33)(37,40);(1,3,5,8)(2,6,4,7)(13,21)(17,22)(18,24)(20,23)(26,31)(29,38)(32,34)(37,40);(1,4,5,2)(3,7,8,6)(9,11)(10,16)(12,15)(13,18)(14,19)(17,23)(20,22)(21,24)(25,27)(26,32)(28,36)(29,37)(30,35)(31,34)(33,39)(38,40);(9,12,14,16)(10,11,15,19)(13,20,21,23)(17,18,22,24)(25,28,30,33)(26,29,31,38)(27,36,35,39)(32,37,34,40);(9,13,14,21)(10,17,15,22)(11,18,19,24)(12,20,16,23)(25,29)(26,33)(27,37)(28,31)(30,38)(32,39)(34,36)(35,40)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=i,e^2=gi,f^2=ghi,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, 8178529);