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 |
| prime radical | #35102 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4496 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,15)(12,20)(13,21)(14,16)(17,22)(18,24)(19,23)(25,26,30,34)(27,35,37,31)(28,38,36,32)(29,33,39,40);(3,7)(6,8)(9,11,14,17)(10,15,16,22)(12,18,21,23)(13,19,20,24)(25,27,30,37)(26,31,34,35)(28,29,36,39)(32,33,38,40);(1,2)(3,6)(4,5)(7,8)(10,16)(11,17)(12,21)(19,24)(28,36)(29,39)(32,38)(33,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,28,30,36)(26,32,34,38)(27,29,37,39)(31,33,35,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,13)(11,18)(14,21)(15,19)(16,20)(17,23)(22,24)(25,29)(26,33)(27,36)(28,37)(30,39)(31,38)(32,35)(34,40);(9,13)(10,12)(11,19)(14,20)(15,18)(16,21)(17,24)(22,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,37)(36,38)(39,40)Polycyclic presentation:
2^9|a^2=g,b^2=gi,d^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=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8033278);