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 | |
4384 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,17)(12,20)(13,15)(14,19)(16,22)(18,23)(21,24)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(2,5)(6,7)(9,11,13,19)(10,14,15,17)(12,18,22,21)(16,23,20,24)(26,30)(31,33)(32,37)(38,39);(1,2)(3,6)(4,5)(7,8)(10,15)(14,17)(16,20)(23,24)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,21)(13,19)(15,17)(16,24)(18,22)(20,23)(25,27,29,35)(26,31,30,33)(28,34,36,40)(32,38,37,39);(1,3)(2,6)(4,8)(5,7)(9,12)(10,16)(11,18)(13,22)(14,23)(15,20)(17,24)(19,21)(25,28,29,36)(26,32,30,37)(27,34,35,40)(31,38,33,39);(9,10)(11,14)(12,16)(13,15)(17,19)(18,23)(20,22)(21,24)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,d^2=gh,e^2=gh,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, 8192711);