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 | |
4366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(25,26,30,33)(27,31,35,29)(28,36,39,32)(34,38,37,40);(1,2,5,6)(3,7,4,8)(9,10)(11,16)(12,17)(13,21)(14,15)(18,22)(19,20)(23,24)(25,27)(26,29)(28,34)(30,35)(31,33)(32,38)(36,40)(37,39);(10,15)(12,16)(18,21)(20,23)(25,27,30,35)(26,29,33,31)(28,37,39,34)(32,40,36,38);(1,3,5,4)(2,7,6,8)(9,11,14,17)(10,16,15,12)(13,19,22,24)(18,23,21,20)(25,27)(26,31)(28,34)(29,33)(30,35)(32,40)(36,38)(37,39);(1,3,5,4)(2,7,6,8)(9,12,14,16)(10,17,15,11)(13,20,22,23)(18,24,21,19)(25,28)(26,32)(27,34)(29,38)(30,39)(31,40)(33,36)(35,37);(1,4,5,3)(2,8,6,7)(9,13)(10,18)(11,19)(12,20)(14,22)(15,21)(16,23)(17,24)(25,29,30,31)(26,27,33,35)(28,38,39,40)(32,34,36,37)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=h,d^2=gi,e^2=gi,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7635445);