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 | |
4330 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)(9,10,14,15)(11,16,20,23)(12,13,17,22)(18,24,21,19)(25,26)(27,29)(28,36)(30,31)(32,34)(33,38)(35,39)(37,40);(2,6)(3,7)(10,15)(12,17)(16,23)(18,21)(26,31)(27,34)(33,36)(35,40);(1,3)(2,4)(5,7)(6,8)(12,17)(13,22)(18,21)(19,24)(25,27,30,34)(26,32,31,29)(28,35,38,40)(33,39,36,37);(1,4)(2,3)(5,8)(6,7)(9,11,14,20)(10,16,15,23)(12,21,17,18)(13,19,22,24)(25,28)(26,33)(27,35)(29,37)(30,38)(31,36)(32,39)(34,40);(9,12,14,17)(10,13,15,22)(11,18,20,21)(16,24,23,19)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(9,13)(10,17)(11,19)(12,15)(14,22)(16,18)(20,24)(21,23)(25,29,30,32)(26,34,31,27)(28,37,38,39)(33,40,36,35)Polycyclic presentation:
2^9|a^2=gh,c^2=h,d^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8947256);