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 | |
4546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,4)(6,7)(9,10)(11,13)(12,20)(14,16)(15,19)(17,22)(18,23)(21,24)(25,26,29,30)(27,33,35,31)(28,32,36,37)(34,39,40,38);(1,2,5,8)(3,6,4,7)(9,11)(10,15)(12,18)(13,16)(14,19)(17,23)(20,21)(22,24)(26,30)(31,33)(32,37)(38,39);(1,3,5,4)(2,6,8,7)(10,16)(13,15)(17,20)(21,23)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40);(1,2)(3,6)(4,7)(5,8)(9,12,14,22)(10,17,16,20)(11,18,19,24)(13,21,15,23)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40);(1,4,5,3)(2,7,8,6)(9,11)(10,15)(12,18)(13,16)(14,19)(17,23)(20,21)(22,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(9,13,14,15)(10,11,16,19)(12,21,22,23)(17,18,20,24)(25,28,29,36)(26,32,30,37)(27,34,35,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=gh,d^2=gi,e^2=gh,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, 8094157);