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 | |
4310 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)(3,4,8,7)(9,10,15,19)(11,16,20,24)(12,14,18,23)(13,22,21,17)(25,26)(27,34)(28,31)(29,36)(30,33)(32,38)(35,37)(39,40);(2,6)(3,8)(11,20)(13,21)(16,24)(17,22)(26,28)(27,33)(32,36)(35,39);(9,11,15,20)(10,16,19,24)(12,21,18,13)(14,17,23,22)(25,27)(26,30)(28,34)(29,35)(31,33)(32,37)(36,40)(38,39);(2,6)(4,7)(9,12)(10,14)(11,21)(13,20)(15,18)(16,17)(19,23)(22,24)(25,28,31,26)(27,34,33,30)(29,36,38,32)(35,40,39,37);(1,3,5,8)(2,7,6,4)(9,13,15,21)(10,17,19,22)(11,12,20,18)(14,16,23,24)(25,29)(26,32)(27,35)(28,36)(30,37)(31,38)(33,39)(34,40);(1,4,5,7)(2,8,6,3)(9,14)(10,18)(11,17)(12,19)(13,16)(15,23)(20,22)(21,24)(25,30,31,34)(26,33,28,27)(29,37,38,40)(32,39,36,35)Polycyclic presentation:
2^9|a^2=gi,c^2=hi,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8400131);