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 | |
4448 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(6,8)(9,10,15,19)(11,20,22,16)(12,13,23,17)(14,24,21,18)(27,33)(31,35)(34,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,14)(13,18)(15,22)(17,24)(19,20)(21,23)(25,26,30,29)(27,31,33,35)(28,36,37,32)(34,40,39,38);(2,5)(6,8)(9,12,15,23)(10,17,19,13)(11,14,22,21)(16,24,20,18)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(1,2)(3,6)(4,5)(7,8)(9,12)(10,17)(11,21)(13,19)(14,22)(15,23)(16,18)(20,24)(25,28,30,37)(26,32,29,36)(27,34,33,39)(31,38,35,40);(1,2)(3,6)(4,5)(7,8)(9,13,15,17)(10,12,19,23)(11,18,22,24)(14,20,21,16)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(1,3)(2,6)(4,7)(5,8)(9,14,15,21)(10,18,19,24)(11,12,22,23)(13,20,17,16)(25,29,30,26)(27,35,33,31)(28,36,37,32)(34,40,39,38)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=g,d^2=gh,e^2=g,f^2=h,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, 8033266);