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 | |
4542 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,7)(3,5,8,6)(25,26)(27,35)(28,32)(29,33)(30,34)(31,36)(37,40)(38,39);(1,3)(2,5)(4,8)(6,7)(9,10)(11,17)(12,20)(13,14)(15,18)(16,21)(19,23)(22,24)(25,26,30,34)(27,35,36,31)(28,29,33,32)(37,38,39,40);(1,3,4,8)(2,6,7,5)(10,14)(11,18)(12,21)(23,24)(27,36)(28,33)(29,32)(31,35);(3,8)(5,6)(9,11)(10,15)(12,22)(13,18)(14,17)(16,24)(19,21)(20,23)(25,27)(26,31)(28,39)(29,38)(30,36)(32,40)(33,37)(34,35);(1,3)(2,5)(4,8)(6,7)(9,12)(10,16)(11,19)(13,21)(14,20)(15,23)(17,24)(18,22)(25,28,30,33)(26,32,34,29)(27,37,36,39)(31,38,35,40);(1,2)(3,5)(4,7)(6,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,21)(22,24)(25,29,30,32)(26,33,34,28)(27,38,36,40)(31,37,35,39)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8965442);