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 | |
4564 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)(9,10,15,13)(11,16,12,18)(14,21,22,17)(19,24,20,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(2,6)(3,7)(9,11,15,12)(10,16,13,18)(14,19,22,20)(17,23,21,24)(25,26,30,34)(27,31,32,28)(29,33,37,40)(35,38,39,36);(1,3)(2,4)(5,7)(6,8)(10,13)(11,12)(14,22)(23,24)(27,32)(28,31)(29,37)(33,40);(1,4)(2,3)(5,8)(6,7)(9,12,15,11)(10,16,13,18)(14,20,22,19)(17,23,21,24)(25,27)(26,31)(28,34)(29,35)(30,32)(33,38)(36,40)(37,39);(9,13,15,10)(11,18,12,16)(14,21,22,17)(19,24,20,23)(25,28,30,31)(26,32,34,27)(29,36,37,38)(33,39,40,35);(9,14)(10,17)(11,19)(12,20)(13,21)(15,22)(16,23)(18,24)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^9|a^2=i,b^2=gi,d^2=h,e^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8961689);