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 | |
4240 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)(11,17)(14,18)(19,23)(22,24)(25,26,30,31)(27,33,35,28)(29,37,38,32)(34,36,40,39);(2,5)(6,7)(9,10)(11,18)(12,20)(13,16)(14,17)(15,21)(19,22)(23,24)(26,31)(28,33)(32,37)(36,39);(1,2)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,17,18)(12,15,21,20)(19,22,23,24)(25,26)(27,28)(29,32)(30,31)(33,35)(34,36)(37,38)(39,40);(1,3)(2,6)(4,8)(5,7)(9,10)(11,14)(12,15)(13,16)(17,18)(19,22)(20,21)(23,24)(25,27)(26,28)(29,34)(30,35)(31,33)(32,36)(37,39)(38,40);(9,11,13,17)(10,14,16,18)(12,19,21,23)(15,22,20,24)(25,28)(26,27)(29,36)(30,33)(31,35)(32,34)(37,40)(38,39);(9,12)(10,15)(11,19)(13,21)(14,22)(16,20)(17,23)(18,24)(25,29,30,38)(26,32,31,37)(27,34,35,40)(28,36,33,39)Polycyclic presentation:
2^9|a^2=h,c^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8664250);