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 | |
4445 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)(10,15)(11,17)(16,20)(19,22)(25,26)(27,32)(28,33)(29,30)(31,34)(35,40)(36,39)(37,38);(1,3,5,8)(2,6,7,4)(9,10,14,15)(11,18,17,13)(12,20,21,16)(19,23,22,24)(27,34)(31,32)(35,36)(39,40);(1,3)(2,4)(5,8)(6,7)(11,17)(12,21)(13,18)(16,20)(25,27,30,34)(26,31,29,32)(28,36,38,35)(33,40,37,39);(1,4)(2,3)(5,6)(7,8)(9,11)(10,13)(12,22)(14,17)(15,18)(16,24)(19,21)(20,23)(25,27)(26,32)(28,36)(29,31)(30,34)(33,39)(35,38)(37,40);(9,12,14,21)(10,16,15,20)(11,19,17,22)(13,23,18,24)(25,28,30,38)(26,33,29,37)(27,35,34,36)(31,39,32,40);(9,13)(10,11)(12,23)(14,18)(15,17)(16,19)(20,22)(21,24)(25,29,30,26)(27,31,34,32)(28,37,38,33)(35,39,36,40)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=h,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, 8970527);