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 | |
4343 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10,14,16)(11,19,20,15)(12,21,22,17)(13,18,23,24)(25,26,31,30)(27,32,34,29)(28,33,37,38)(35,39,40,36);(3,7)(6,8)(9,11,14,20)(10,15,16,19)(12,18,22,24)(13,17,23,21)(25,27,31,34)(26,32,30,29)(28,35,37,40)(33,39,38,36);(1,2,5,4)(3,6,7,8)(10,16)(12,22)(15,19)(18,24)(27,34)(29,32)(35,40)(36,39);(1,3,5,7)(2,6,4,8)(9,11)(10,15)(12,18)(13,21)(14,20)(16,19)(17,23)(22,24)(25,28)(26,33)(27,35)(29,36)(30,38)(31,37)(32,39)(34,40);(1,4,5,2)(3,8,7,6)(9,12,14,22)(10,17,16,21)(11,18,20,24)(13,15,23,19)(25,29)(26,27)(28,36)(30,34)(31,32)(33,35)(37,39)(38,40);(9,13,14,23)(10,18,16,24)(11,17,20,21)(12,15,22,19)(25,30,31,26)(27,29,34,32)(28,33,37,38)(35,39,40,36)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=gh,d^2=gh,e^2=h,f^2=gi,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, 8458164);