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 | |
4505 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)(3,5)(4,7)(6,8)(9,10,14,17)(11,18,13,15)(12,16,20,23)(19,24,21,22)(25,26,30,32)(27,35,36,31)(28,37,39,33)(29,34,38,40);(2,3)(6,7)(9,11,14,13)(10,15,17,18)(12,19,20,21)(16,22,23,24)(25,27,30,36)(26,31,32,35)(28,38,39,29)(33,40,37,34);(1,2,5,3)(4,6,8,7)(12,20)(16,23)(19,21)(22,24)(26,32)(31,35)(33,37)(34,40);(1,3,5,2)(4,7,8,6)(9,10)(11,15)(12,16)(13,18)(14,17)(19,22)(20,23)(21,24)(25,28,30,39)(26,33,32,37)(27,29,36,38)(31,34,35,40);(1,4)(2,6)(3,7)(5,8)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,26)(27,31)(28,33)(29,34)(30,32)(35,36)(37,39)(38,40);(9,13,14,11)(10,15,17,18)(12,21,20,19)(16,22,23,24)(25,29)(26,34)(27,28)(30,38)(31,33)(32,40)(35,37)(36,39)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=i,d^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8440556);