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 | |
4651 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10,13,14)(11,17,18,15)(12,16,21,22)(19,24,20,23)(25,26,31,34)(27,29,32,37)(28,30,33,39)(35,36,38,40);(3,7)(6,8)(10,14)(11,18)(16,22)(19,20)(25,27,31,32)(26,29,34,37)(28,35,33,38)(30,36,39,40);(1,2,5,4)(3,6,7,8)(10,14)(15,17)(16,22)(23,24)(27,32)(29,37)(35,38)(36,40);(1,3,5,7)(2,6,4,8)(9,11)(10,15)(12,20)(13,18)(14,17)(16,24)(19,21)(22,23)(25,28)(26,30)(27,35)(29,36)(31,33)(32,38)(34,39)(37,40);(1,4,5,2)(3,8,7,6)(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,24)(21,22)(25,29,31,37)(26,32,34,27)(28,36,33,40)(30,38,39,35);(9,12,13,21)(10,16,14,22)(11,19,18,20)(15,23,17,24)(25,30)(26,33)(27,36)(28,34)(29,38)(31,39)(32,40)(35,37)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=gh,d^2=gh,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8528960);