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 | |
4531 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)(9,10,14,15)(11,16,17,12)(13,21,22,18)(19,20,24,23)(25,26,31,34)(27,32,35,40)(28,38,37,30)(29,39,36,33);(1,3,5,8)(2,6,7,4)(10,15)(11,17)(18,21)(19,24)(25,27)(26,32)(28,29)(30,33)(31,35)(34,40)(36,37)(38,39);(1,3)(2,4)(5,8)(6,7)(10,15)(11,17)(13,22)(20,23)(27,35)(28,37)(30,38)(32,40);(1,4)(2,3)(5,6)(7,8)(9,11)(10,16)(12,15)(13,19)(14,17)(18,23)(20,21)(22,24)(25,28)(26,30)(27,36)(29,35)(31,37)(32,39)(33,40)(34,38);(9,12,14,16)(10,17,15,11)(13,20,22,23)(18,24,21,19)(25,29,31,36)(26,33,34,39)(27,37,35,28)(30,32,38,40);(9,13,14,22)(10,18,15,21)(11,19,17,24)(12,20,16,23)(25,30)(26,28)(27,33)(29,32)(31,38)(34,37)(35,39)(36,40)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8842032);