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 | |
4728 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,7,5,8)(9,10)(11,17)(12,18)(13,16)(14,15)(19,24)(20,21)(22,23)(25,26)(27,32)(28,34)(29,39)(30,35)(31,33)(36,38)(37,40);(1,3,6,5)(2,7,4,8)(10,15)(11,18)(16,23)(19,21)(25,27)(26,32)(28,33)(29,36)(30,37)(31,34)(35,40)(38,39);(1,3,6,5)(2,8,4,7)(10,15)(12,17)(16,23)(20,24)(26,33)(27,34)(29,35)(37,38);(1,4,6,2)(3,7,5,8)(9,11)(10,12)(13,21)(14,18)(15,17)(16,24)(19,22)(20,23)(25,28,31,32)(26,34,33,27)(29,38,35,37)(30,40,39,36);(1,5,6,3)(2,7,4,8)(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,23)(22,24)(25,29,31,35)(26,30,33,39)(27,36,34,40)(28,38,32,37);(9,13,14,22)(10,16,15,23)(11,19,18,21)(12,20,17,24)(25,30)(26,35)(27,37)(28,36)(29,33)(31,39)(32,40)(34,38)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=ghi,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8440324);