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 | |
4554 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,18)(12,16)(13,21)(14,15)(17,22)(19,20)(23,24)(25,26,30,33)(27,28,31,36)(29,32,38,40)(34,35,37,39);(2,5)(6,7)(10,15)(11,16)(13,22)(20,24)(25,27,30,31)(26,28,33,36)(29,37,38,34)(32,39,40,35);(1,2)(3,6)(4,5)(7,8)(10,15)(12,18)(17,21)(20,24)(25,27)(26,28)(29,34)(30,31)(32,35)(33,36)(37,38)(39,40);(1,3,4,8)(2,6,5,7)(9,11,14,16)(10,12,15,18)(13,19,22,23)(17,20,21,24)(25,27)(26,28)(29,34)(30,31)(32,35)(33,36)(37,38)(39,40);(9,12)(10,16)(11,15)(13,20)(14,18)(17,23)(19,21)(22,24)(25,28)(26,31)(27,33)(29,35)(30,36)(32,37)(34,40)(38,39);(9,13)(10,17)(11,19)(12,20)(14,22)(15,21)(16,23)(18,24)(25,29)(26,32)(27,34)(28,35)(30,38)(31,37)(33,40)(36,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10239368);