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 | |
4699 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(3,5)(9,10)(11,16)(12,18)(13,22)(14,21)(15,17)(19,23)(20,24)(26,29)(27,33)(32,37)(34,36);(1,2,6,4)(3,8,5,7)(9,11)(10,16)(12,15)(13,21)(14,22)(17,18)(19,24)(20,23)(25,26,30,29)(27,31,33,35)(28,32,38,37)(34,39,36,40);(1,2,6,4)(3,8,5,7)(10,17)(11,12)(19,22)(20,21)(27,33)(31,35)(34,36)(39,40);(1,3,6,5)(2,7,4,8)(9,12,15,11)(10,18,17,16)(13,21,23,20)(14,19,24,22)(25,27)(26,31)(28,36)(29,35)(30,33)(32,40)(34,38)(37,39);(1,4,6,2)(3,7,5,8)(9,13)(10,19)(11,21)(12,20)(14,18)(15,23)(16,24)(17,22)(25,28,30,38)(26,32,29,37)(27,34,33,36)(31,39,35,40);(1,5,6,3)(2,7,4,8)(9,14)(10,20)(11,22)(12,19)(13,18)(15,24)(16,23)(17,21)(25,29,30,26)(27,35,33,31)(28,37,38,32)(34,40,36,39)Polycyclic presentation:
2^9|b^2=g,c^2=hi,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8064211);