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 | |
4570 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)(11,19)(12,23)(13,18)(14,15)(16,22)(17,21)(20,24)(26,31)(27,33)(32,36)(35,38);(1,3,5,8)(2,6,7,4)(9,11,15,22)(10,16,14,19)(12,13,21,20)(17,18,23,24)(25,26,30,31)(27,34,33,29)(28,36,37,32)(35,39,38,40);(1,3)(2,4)(5,8)(6,7)(10,14)(12,21)(13,20)(16,19)(27,33)(28,37)(29,34)(32,36);(1,4)(2,3)(5,6)(7,8)(9,12)(10,17)(11,20)(13,22)(14,23)(15,21)(16,24)(18,19)(25,27)(26,29)(28,38)(30,33)(31,34)(32,40)(35,37)(36,39);(9,13,15,20)(10,18,14,24)(11,21,22,12)(16,23,19,17)(25,28,30,37)(26,32,31,36)(27,35,33,38)(29,39,34,40);(9,14,15,10)(11,19,22,16)(12,23,21,17)(13,24,20,18)(25,29)(26,27)(28,39)(30,34)(31,33)(32,35)(36,38)(37,40)Polycyclic presentation:
2^9|a^2=i,b^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,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, 8970471);