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 | |
4438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,8)(4,6)(9,10,14,18)(11,15,19,23)(12,21,22,16)(13,20,24,17)(25,26,30,33)(27,31,35,29)(28,36,39,32)(34,38,37,40);(1,2)(3,4)(5,7)(6,8)(9,11)(10,15)(12,17)(13,21)(14,19)(16,24)(18,23)(20,22)(25,27,30,35)(26,29,33,31)(28,37,39,34)(32,40,36,38);(1,3)(2,6)(4,7)(5,8)(11,19)(12,22)(15,23)(16,21)(25,27,30,35)(26,31,33,29)(28,34,39,37)(32,40,36,38);(1,3,5,8)(2,6,7,4)(9,12)(10,16)(11,20)(13,15)(14,22)(17,19)(18,21)(23,24)(25,27)(26,31)(28,37)(29,33)(30,35)(32,38)(34,39)(36,40);(1,3)(2,4)(5,8)(6,7)(9,13)(10,17)(11,16)(12,23)(14,24)(15,22)(18,20)(19,21)(25,28)(26,32)(27,34)(29,38)(30,39)(31,40)(33,36)(35,37);(1,4)(2,3)(5,6)(7,8)(9,11)(10,15)(12,20)(13,16)(14,19)(17,22)(18,23)(21,24)(25,29,30,31)(26,27,33,35)(28,38,39,40)(32,34,36,37)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=g,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8841060);