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 | |
4367 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)(3,4)(5,6)(7,8)(25,26)(27,30)(28,36)(29,32)(31,37)(33,35)(34,39)(38,40);(2,6)(4,7)(9,10,14,15)(11,16,19,24)(12,17,22,21)(13,18,23,20)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,37)(38,40);(1,2,5,6)(3,7,8,4)(10,15)(11,19)(12,22)(13,23)(25,26,29,32)(27,33,35,30)(28,36,37,31)(34,38,40,39);(1,3)(2,7)(4,6)(5,8)(9,11)(10,16)(12,18)(13,21)(14,19)(15,24)(17,23)(20,22)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,37)(38,40);(1,4,5,7)(2,3,6,8)(9,12)(10,17)(11,20)(13,16)(14,22)(15,21)(18,19)(23,24)(25,27,29,35)(26,30,32,33)(28,34,37,40)(31,38,36,39);(1,3)(2,7)(4,6)(5,8)(9,13)(10,18)(11,21)(12,16)(14,23)(15,20)(17,19)(22,24)(25,28,29,37)(26,31,32,36)(27,34,35,40)(30,38,33,39)Polycyclic presentation:
2^9|b^2=hi,c^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8886640);