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 | |
4543 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)(11,18)(12,17)(13,16)(14,19)(15,21)(20,23)(22,24)(28,36)(31,37)(35,39)(38,40);(1,3,5,8)(2,6,7,4)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,24)(18,19)(25,26)(27,34)(28,37)(29,32)(30,33)(31,36)(35,38)(39,40);(1,3)(2,4)(5,8)(6,7)(9,12,14,13)(10,16,19,17)(11,20,21,22)(15,23,18,24)(25,27,30,32)(26,29,33,34)(28,35,36,39)(31,38,37,40);(1,4)(2,3)(5,6)(7,8)(9,13,14,12)(10,17,19,16)(11,20,21,22)(15,23,18,24)(25,26)(27,29)(28,31)(30,33)(32,34)(35,38)(36,37)(39,40);(9,10)(11,18)(12,16)(13,17)(14,19)(15,21)(20,24)(22,23)(25,28)(26,31)(27,35)(29,38)(30,36)(32,39)(33,37)(34,40);(9,11)(10,18)(12,20)(13,22)(14,21)(15,19)(16,24)(17,23)(25,29)(26,32)(27,33)(28,38)(30,34)(31,39)(35,37)(36,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=gi,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8386734);