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 | |
4272 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,18)(11,15,21,24)(12,13,17,16)(19,20,23,22)(26,32)(27,35)(28,29)(38,39);(1,2)(3,7)(4,5)(6,8)(9,11,14,21)(10,15,18,24)(12,19,17,23)(13,20,16,22)(25,26,31,32)(27,30,35,40)(28,34,29,33)(36,39,37,38);(2,5)(6,7)(9,12,14,17)(10,16,18,13)(11,19,21,23)(15,22,24,20)(25,27)(26,30)(28,36)(29,37)(31,35)(32,40)(33,38)(34,39);(9,11)(10,15)(12,19)(13,20)(14,21)(16,22)(17,23)(18,24)(25,28,31,29)(26,33,32,34)(27,36,35,37)(30,38,40,39);(1,3,4,8)(2,6,5,7)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,29,31,28)(26,34,32,33)(27,37,35,36)(30,39,40,38);(1,2)(3,6)(4,5)(7,8)(9,13,14,16)(10,17,18,12)(11,20,21,22)(15,23,24,19)(25,30)(26,35)(27,32)(28,38)(29,39)(31,40)(33,37)(34,36)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=gh,d^2=g,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8122414);