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 | |
4691 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)(11,13)(12,17)(14,22)(15,16)(18,23)(19,24)(20,21)(26,30)(27,33)(32,38)(34,35);(3,7)(6,8)(9,11)(10,13)(12,15)(14,19)(16,17)(18,21)(20,23)(22,24)(25,26)(27,31)(28,32)(29,30)(33,37)(34,39)(35,40)(36,38);(1,2,5,4)(3,6,7,8)(10,16)(11,12)(18,22)(19,20)(27,33)(31,37)(34,35)(39,40);(1,3,5,7)(2,6,4,8)(9,12,15,11)(10,17,16,13)(14,20,23,19)(18,24,22,21)(25,27)(26,31)(28,35)(29,33)(30,37)(32,40)(34,36)(38,39);(1,4,5,2)(3,8,7,6)(9,13,15,17)(10,12,16,11)(14,21,23,24)(18,20,22,19)(25,28,29,36)(26,32,30,38)(27,34,33,35)(31,39,37,40);(9,14)(10,18)(11,19)(12,20)(13,21)(15,23)(16,22)(17,24)(25,26)(27,31)(28,32)(29,30)(33,37)(34,39)(35,40)(36,38)Polycyclic presentation:
2^9|c^2=hi,d^2=h,e^2=gh,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, 8064517);