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 | |
4512 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)(11,19)(12,21)(13,23)(14,15)(16,20)(17,22)(18,24)(25,26)(27,33)(28,36)(29,30)(31,35)(32,37)(34,38)(39,40);(1,2,5,4)(3,7,8,6)(10,15)(13,24)(16,19)(17,21)(25,27,30,35)(26,31,29,33)(28,34,37,40)(32,38,36,39);(1,3,5,8)(2,6,4,7)(10,15)(11,20)(17,21)(18,23)(28,37)(32,36)(34,40)(38,39);(1,4,5,2)(3,7,8,6)(9,11)(10,16)(12,18)(13,21)(14,20)(15,19)(17,24)(22,23)(25,28,30,37)(26,32,29,36)(27,34,35,40)(31,38,33,39);(9,12)(10,17)(11,18)(13,19)(14,22)(15,21)(16,24)(20,23)(25,29,30,26)(27,31,35,33)(28,36,37,32)(34,38,40,39);(9,13,14,24)(10,18,15,23)(11,21,20,17)(12,19,22,16)(25,27)(26,33)(28,34)(29,31)(30,35)(32,39)(36,38)(37,40)Polycyclic presentation:
2^9|b^2=gh,c^2=i,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8664411);