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 | |
4548 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)(3,7)(4,6)(5,8)(9,10,14,18)(11,12,16,15)(13,17,21,24)(19,20,23,22)(25,26,31,35)(27,36,37,32)(28,33,38,30)(29,39,40,34);(1,2,4,6)(3,5,8,7)(9,11,14,16)(10,15,18,12)(13,19,21,23)(17,22,24,20)(27,37)(29,40)(32,36)(34,39);(1,2)(3,5)(4,6)(7,8)(13,21)(17,24)(19,23)(20,22)(25,27,31,37)(26,32,35,36)(28,29,38,40)(30,39,33,34);(1,3)(2,5)(4,8)(6,7)(9,10)(11,12)(13,17)(14,18)(15,16)(19,20)(21,24)(22,23)(25,28)(26,33)(27,29)(30,35)(31,38)(32,34)(36,39)(37,40);(9,12)(10,16)(11,18)(13,20)(14,15)(17,23)(19,24)(21,22)(25,29)(26,34)(27,38)(28,37)(30,32)(31,40)(33,36)(35,39);(9,13,14,21)(10,17,18,24)(11,19,16,23)(12,20,15,22)(25,30,31,33)(26,28,35,38)(27,34,37,39)(29,32,40,36)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8306672);