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 | |
4645 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10,13,16)(11,17,19,14)(12,15,21,20)(18,23,24,22)(25,26)(27,32)(28,39)(29,33)(30,34)(31,35)(36,37)(38,40);(2,6)(3,7)(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,21)(23,24)(25,27,31,38)(26,32,35,40)(28,29,34,36)(30,37,39,33);(1,3)(2,4)(5,7)(6,8)(25,28,31,34)(26,30,35,39)(27,36,38,29)(32,33,40,37);(1,4)(2,3)(5,8)(6,7)(9,10)(11,14)(12,20)(13,16)(15,21)(17,19)(18,23)(22,24)(25,27)(26,32)(28,36)(29,34)(30,33)(31,38)(35,40)(37,39);(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(25,29,31,36)(26,33,35,37)(27,34,38,28)(30,40,39,32);(9,12)(10,15)(11,18)(13,21)(14,22)(16,20)(17,23)(19,24)(25,30)(26,34)(27,37)(28,35)(29,40)(31,39)(32,36)(33,38)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8345238);