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 | |
4640 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(6,8)(9,10,14,13)(11,17,19,15)(12,20,21,16)(18,22,24,23)(25,26,30,34)(27,35,37,31)(28,38,39,32)(29,36,40,33);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,23)(21,24)(25,27)(26,31)(28,33)(29,32)(30,37)(34,35)(36,39)(38,40);(2,5)(6,8)(12,21)(16,20)(18,24)(22,23)(25,28,30,39)(26,32,34,38)(27,33,37,36)(29,35,40,31);(1,2)(3,6)(4,5)(7,8)(9,12,14,21)(10,16,13,20)(11,18,19,24)(15,22,17,23)(25,28)(26,32)(27,36)(29,35)(30,39)(31,40)(33,37)(34,38);(9,13,14,10)(11,17,19,15)(12,20,21,16)(18,23,24,22)(25,29,30,40)(26,33,34,36)(27,32,37,38)(28,35,39,31);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,23)(21,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,37)(36,40)(38,39)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8013275);