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 | |
4505 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)(3,5)(4,7)(6,8)(9,10)(11,17)(12,15)(13,16)(14,20)(18,22)(19,23)(21,24)(25,26)(27,32)(28,30)(29,38)(31,33)(34,37)(35,40)(36,39);(1,2,6,8)(3,5,7,4)(9,11,15,13)(10,16,12,17)(14,21,22,19)(18,24,20,23)(25,27)(26,32)(28,37)(29,35)(30,34)(31,36)(33,39)(38,40);(1,3,6,7)(2,4,8,5)(10,12)(16,17)(18,20)(23,24)(26,33)(27,36)(28,35)(34,38);(1,4,6,5)(2,3,8,7)(9,12,15,10)(11,17,13,16)(14,20,22,18)(19,24,21,23)(25,28,31,35)(26,30,33,40)(27,37,36,29)(32,34,39,38);(1,2)(3,4)(5,7)(6,8)(9,13,15,11)(10,17,12,16)(14,21,22,19)(18,24,20,23)(25,29,31,37)(26,34,33,38)(27,35,36,28)(30,39,40,32);(1,5,6,4)(2,7,8,3)(9,14,15,22)(10,18,12,20)(11,19,13,21)(16,23,17,24)(25,30)(26,35)(27,38)(28,33)(29,39)(31,40)(32,37)(34,36)Polycyclic presentation:
2^9|b^2=g,c^2=ghi,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8664208);