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 | |
4561 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)(11,19)(12,21)(13,18)(14,15)(16,20)(17,22)(23,24)(25,26,29,33)(27,31,35,30)(28,36,37,32)(34,38,40,39);(2,6)(3,7)(10,15)(12,22)(16,19)(18,23)(25,27)(26,30)(28,34)(29,35)(31,33)(32,38)(36,39)(37,40);(1,3,5,7)(2,4,6,8)(25,26)(27,31)(28,32)(29,33)(30,35)(34,39)(36,37)(38,40);(1,4)(2,7)(3,6)(5,8)(9,11)(10,16)(12,23)(13,21)(14,20)(15,19)(17,24)(18,22)(25,26)(27,30)(28,32)(29,33)(31,35)(34,38)(36,37)(39,40);(9,12)(10,17)(11,18)(13,16)(14,22)(15,21)(19,24)(20,23)(25,27)(26,31)(28,34)(29,35)(30,33)(32,39)(36,38)(37,40);(9,13)(10,18)(11,17)(12,16)(14,24)(15,23)(19,22)(20,21)(25,28,29,37)(26,32,33,36)(27,34,35,40)(30,38,31,39)Polycyclic presentation:
2^9|a^2=h,c^2=hi,f^2=h,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, 10331861);