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 | |
4524 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,18)(12,15)(13,16)(14,17)(19,24)(20,22)(21,23);(1,3)(2,6)(4,7)(5,8)(9,11,15,13)(10,16,12,18)(14,19,22,21)(17,23,20,24)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(2,4)(6,7)(14,22)(17,20)(19,21)(23,24)(26,30)(31,33)(32,37)(38,39);(1,3)(2,6)(4,7)(5,8)(9,12,15,10)(11,16,13,18)(14,20,22,17)(19,23,21,24)(25,27,29,35)(26,31,30,33)(28,34,36,40)(32,38,37,39);(1,3)(2,6)(4,7)(5,8)(9,13,15,11)(10,16,12,18)(14,21,22,19)(17,23,20,24)(25,28,29,36)(26,32,30,37)(27,34,35,40)(31,38,33,39);(1,4,5,2)(3,7,8,6)(9,14,15,22)(10,17,12,20)(11,19,13,21)(16,23,18,24)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40)Polycyclic presentation:
2^9|a^2=h,b^2=gh,d^2=ghi,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8100950);