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 | |
4483 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)(3,4)(5,6)(7,8)(9,10)(11,12)(13,16)(14,17)(15,18)(19,20)(21,22)(23,24)(25,26)(27,32)(28,34)(29,30)(31,35)(33,39)(36,38)(37,40);(2,6)(3,7)(9,11)(10,12)(13,19)(14,15)(16,20)(17,18)(21,24)(22,23)(25,27,31,30)(26,32,35,29)(28,37,39,36)(33,38,34,40);(1,3,5,7)(2,4,6,8)(9,10)(11,18)(12,15)(13,16)(14,17)(19,24)(20,23)(21,22)(25,28)(26,33)(27,36)(29,38)(30,37)(31,39)(32,40)(34,35);(1,4)(2,7)(3,6)(5,8)(9,10)(11,12)(13,21)(14,17)(15,18)(16,22)(19,24)(20,23)(25,28)(26,34)(27,37)(29,38)(30,36)(31,39)(32,40)(33,35);(9,12)(10,15)(11,17)(13,20)(14,18)(16,23)(19,21)(22,24)(25,29,31,32)(26,27,35,30)(28,38,39,40)(33,36,34,37);(9,13,14,22)(10,16,17,21)(11,19,15,23)(12,20,18,24)(25,30,31,27)(26,32,35,29)(28,37,39,36)(33,40,34,38)Polycyclic presentation:
2^9|b^2=h,c^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10353681);