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 | |
4530 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,8)(4,6)(9,10,14,15)(11,12,17,16)(13,21,22,18)(19,23,24,20)(25,26,30,31)(27,34,35,32)(28,29,33,39)(36,37,38,40);(1,2)(3,4)(5,7)(6,8)(10,15)(11,17)(18,21)(19,24)(26,31)(27,35)(28,33)(37,40);(1,3)(2,6)(4,7)(5,8)(10,15)(12,16)(13,22)(19,24)(28,33)(29,39)(36,38)(37,40);(1,2,5,7)(3,6,8,4)(9,11)(10,16)(12,15)(13,19)(14,17)(18,23)(20,21)(22,24)(25,27,30,35)(26,32,31,34)(28,38,33,36)(29,37,39,40);(1,2)(3,4)(5,7)(6,8)(9,12,14,16)(10,17,15,11)(13,20,22,23)(18,24,21,19)(25,28,30,33)(26,29,31,39)(27,36,35,38)(32,40,34,37);(1,4)(2,3)(5,6)(7,8)(9,13)(10,18)(11,19)(12,20)(14,22)(15,21)(16,23)(17,24)(25,29)(26,33)(27,37)(28,31)(30,39)(32,36)(34,38)(35,40)Polycyclic presentation:
2^9|a^2=g,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,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, 8947617);