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 | |
4650 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)(6,8)(9,10)(11,16)(12,14)(13,22)(15,17)(18,24)(19,23)(20,21)(26,31)(27,33)(29,37)(36,40);(3,7)(6,8)(9,11,15,13)(10,16,17,22)(12,21,18,19)(14,20,24,23)(25,26)(27,28)(29,32)(30,31)(33,35)(34,36)(37,38)(39,40);(1,2,5,4)(3,6,7,8)(10,17)(11,13)(12,18)(20,23)(27,33)(28,35)(34,39)(36,40);(1,3,5,7)(2,6,4,8)(9,12,15,18)(10,14,17,24)(11,19,13,21)(16,23,22,20)(25,27)(26,28)(29,34)(30,33)(31,35)(32,36)(37,39)(38,40);(1,4,5,2)(3,8,7,6)(9,13,15,11)(10,16,17,22)(12,21,18,19)(14,23,24,20)(25,28)(26,27)(29,36)(30,35)(31,33)(32,34)(37,40)(38,39);(9,14)(10,18)(11,20)(12,17)(13,23)(15,24)(16,19)(21,22)(25,29,30,37)(26,32,31,38)(27,34,33,39)(28,36,35,40)Polycyclic presentation:
2^9|b^2=i,c^2=hi,d^2=h,e^2=h,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^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8034702);