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 | |
4585 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,19)(12,21)(13,18)(14,15)(16,20)(17,23)(22,24)(26,31)(28,35)(33,37)(36,40);(1,2)(3,6)(4,5)(7,8)(10,15)(11,20)(17,21)(18,22)(26,31)(27,32)(33,37)(34,39);(2,5)(6,8)(10,15)(13,24)(16,19)(17,21)(25,26,30,31)(27,28,32,35)(29,37,38,33)(34,40,39,36);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,22)(13,21)(14,20)(15,19)(17,24)(18,23)(25,27)(26,28)(29,34)(30,32)(31,35)(33,36)(37,40)(38,39);(1,3)(2,6)(4,7)(5,8)(9,12,14,23)(10,17,15,21)(11,18,20,22)(13,19,24,16)(25,28)(26,32)(27,31)(29,36)(30,35)(33,39)(34,37)(38,40);(1,2)(3,6)(4,5)(7,8)(9,13)(10,18)(11,21)(12,19)(14,24)(15,22)(16,23)(17,20)(25,29)(26,33)(27,34)(28,36)(30,38)(31,37)(32,39)(35,40)Polycyclic presentation:
2^9|c^2=gh,d^2=ghi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8195597);