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 | |
4669 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,17)(12,18)(13,16)(14,15)(19,22)(20,21)(23,24)(25,26)(27,32)(28,36)(29,33)(30,35)(31,37)(34,38)(39,40);(1,3,5,8)(2,6,7,4)(10,15)(11,18)(16,24)(19,21)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,39)(37,40);(1,3)(2,4)(5,8)(6,7)(10,15)(11,18)(16,24)(19,21)(28,37)(31,36)(34,40)(38,39);(1,4)(2,3)(5,6)(7,8)(9,11)(10,12)(13,21)(14,18)(15,17)(16,22)(19,23)(20,24)(25,28)(26,31)(27,34)(29,37)(30,38)(32,39)(33,36)(35,40);(9,12)(10,11)(13,22)(14,17)(15,18)(16,21)(19,24)(20,23)(25,27)(26,32)(28,34)(29,35)(30,33)(31,39)(36,38)(37,40);(9,13,14,23)(10,16,15,24)(11,19,18,21)(12,20,17,22)(25,26)(27,30)(28,36)(29,33)(31,37)(32,35)(34,39)(38,40)Polycyclic presentation:
2^9|a^2=i,b^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8620098);