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 | |
4488 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)(9,10,15,18)(11,12,13,16)(14,17,22,24)(19,20,21,23)(25,26)(27,31)(28,35)(29,30)(32,36)(33,39)(34,37)(38,40);(1,3)(2,6)(4,7)(5,8)(9,11,15,13)(10,16,18,12)(14,19,22,21)(17,23,24,20)(25,26)(27,31)(28,32)(29,30)(33,38)(34,37)(35,36)(39,40);(2,4)(6,7)(14,22)(17,24)(19,21)(20,23)(26,30)(31,37)(32,35)(38,39);(1,3)(2,6)(4,7)(5,8)(9,12,15,16)(10,11,18,13)(14,20,22,23)(17,19,24,21)(25,27,29,34)(26,31,30,37)(28,33,36,40)(32,38,35,39);(1,3)(2,6)(4,7)(5,8)(9,13,15,11)(10,16,18,12)(14,21,22,19)(17,23,24,20)(25,28)(26,32)(27,33)(29,36)(30,35)(31,38)(34,40)(37,39);(1,4,5,2)(3,7,8,6)(9,14,15,22)(10,17,18,24)(11,19,13,21)(12,20,16,23)(25,26)(27,31)(28,32)(29,30)(33,38)(34,37)(35,36)(39,40)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,d^2=gh,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, 8099751);