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 | |
4406 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,8,4,7)(13,21)(16,20)(18,24)(22,23)(27,28)(29,37)(31,32)(33,40);(2,6)(7,8)(10,12)(15,17)(16,20)(22,23)(25,26,30,34)(27,32,28,31)(29,33,37,40)(35,39,36,38);(1,3,5,4)(2,7,6,8)(9,10,14,12)(11,15,19,17)(13,16,21,20)(18,22,24,23)(25,27,30,28)(26,31,34,32)(29,35,37,36)(33,38,40,39);(1,2)(3,7)(4,8)(5,6)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(1,2)(3,7)(4,8)(5,6)(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22)(25,28,30,27)(26,32,34,31)(29,36,37,35)(33,39,40,38);(1,4,5,3)(2,8,6,7)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=gi,d^2=ghi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8017692);