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 | |
4524 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(3,7)(9,10)(11,16)(12,17)(13,21)(14,15)(18,22)(19,20)(23,24)(25,26)(27,32)(28,31)(29,33)(30,38)(34,40)(35,36)(37,39);(1,2,6,5)(3,8,7,4)(9,11)(10,12)(13,19)(14,17)(15,16)(18,20)(21,23)(22,24)(25,27)(26,32)(28,35)(29,30)(31,36)(33,38)(34,37)(39,40);(1,3)(2,4)(5,8)(6,7)(10,15)(12,16)(13,22)(19,24)(25,26,31,28)(27,35,36,32)(29,37,39,33)(30,38,40,34);(1,2,6,5)(3,4,7,8)(9,11)(10,16)(12,15)(13,19)(14,17)(18,23)(20,21)(22,24)(25,28,31,26)(27,35,36,32)(29,33,39,37)(30,38,40,34);(1,4,6,8)(2,7,5,3)(9,12,14,16)(10,17,15,11)(13,20,22,23)(18,24,21,19)(25,29)(26,33)(27,30)(28,37)(31,39)(32,38)(34,35)(36,40);(1,5,6,2)(3,8,7,4)(9,13,14,22)(10,18,15,21)(11,19,17,24)(12,20,16,23)(25,30)(26,34)(27,29)(28,38)(31,40)(32,37)(33,35)(36,39)Polycyclic presentation:
2^9|b^2=ghi,c^2=gh,d^2=i,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8946457);