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 | |
4651 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(10,15)(11,16)(17,22)(18,23)(26,31)(28,35)(33,37)(36,40);(1,3,5,8)(2,6,7,4)(9,10,14,15)(11,12,16,20)(13,17,21,22)(18,19,23,24)(26,31)(27,32)(33,37)(34,39);(1,3)(2,4)(5,8)(6,7)(13,21)(17,22)(18,23)(19,24)(25,26,30,31)(27,28,32,35)(29,37,38,33)(34,40,39,36);(1,4)(2,3)(5,6)(7,8)(9,11)(10,12)(13,18)(14,16)(15,20)(17,19)(21,23)(22,24)(25,27)(26,28)(29,34)(30,32)(31,35)(33,36)(37,40)(38,39);(9,12)(10,16)(11,15)(13,19)(14,20)(17,23)(18,22)(21,24)(25,28)(26,32)(27,31)(29,36)(30,35)(33,39)(34,37)(38,40);(9,13,14,21)(10,17,15,22)(11,18,16,23)(12,19,20,24)(25,29,30,38)(26,33,31,37)(27,34,32,39)(28,36,35,40)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8230522);