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:
(2,6)(7,8)(25,26,29,33)(27,31,35,30)(28,32,37,40)(34,39,36,38);(1,2,5,6)(3,7,4,8)(9,10)(11,16)(12,13)(14,15)(17,22)(18,24)(19,21)(20,23)(25,27)(26,30)(28,36)(29,35)(31,33)(32,39)(34,37)(38,40);(10,15)(13,22)(16,23)(19,24)(25,27,29,35)(26,30,33,31)(28,36,37,34)(32,39,40,38);(9,11)(10,16)(12,21)(13,19)(14,20)(15,23)(17,18)(22,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,39)(36,37)(38,40);(1,3,5,4)(2,7,6,8)(9,12,14,17)(10,13,15,22)(11,18,20,21)(16,24,23,19)(25,26)(27,30)(28,32)(29,33)(31,35)(34,38)(36,39)(37,40);(1,4,5,3)(2,8,6,7)(9,13)(10,17)(11,19)(12,15)(14,22)(16,18)(20,24)(21,23)(25,28,29,37)(26,32,33,40)(27,34,35,36)(30,38,31,39)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7733899);