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 | |
4308 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)(3,8)(4,6)(5,7)(9,10,14,12)(11,15,17,18)(13,16,21,20)(19,22,23,24)(25,26)(27,35)(28,33)(29,34)(30,31)(32,38)(36,40)(37,39);(1,2,6,4)(3,8,5,7)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(26,31)(32,35)(33,34)(39,40);(1,2,6,4)(3,7,5,8)(11,17)(15,18)(19,23)(22,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,38)(36,39)(37,40);(1,3,6,5)(2,7,4,8)(9,12,14,10)(11,18,17,15)(13,16,21,20)(19,22,23,24)(25,27)(26,32)(28,36)(29,37)(30,38)(31,35)(33,39)(34,40);(1,4,6,2)(3,7,5,8)(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,23)(18,24)(25,28,30,29)(26,33,31,34)(27,36,38,37)(32,39,35,40);(1,5,6,3)(2,7,4,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,29,30,28)(26,34,31,33)(27,37,38,36)(32,40,35,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=h,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8198862);