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 | |
4767 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10,14,15)(11,16,19,24)(12,17,22,21)(13,18,23,20)(25,26)(27,32)(28,29)(30,38)(31,33)(34,39)(35,37)(36,40);(2,6)(3,7)(10,15)(12,22)(16,24)(18,20)(25,27,31,28)(26,32,33,29)(30,35,39,36)(34,40,38,37);(1,3)(2,4)(5,7)(6,8)(11,19)(13,23)(16,24)(18,20)(26,33)(27,28)(30,39)(37,40);(1,4)(2,3)(5,8)(6,7)(9,11)(10,16)(12,18)(13,21)(14,19)(15,24)(17,23)(20,22)(25,28,31,27)(26,32,33,29)(30,36,39,35)(34,40,38,37);(9,12)(10,17)(11,20)(13,16)(14,22)(15,21)(18,19)(23,24)(25,29,31,32)(26,28,33,27)(30,37,39,40)(34,36,38,35);(9,13)(10,18)(11,21)(12,16)(14,23)(15,20)(17,19)(22,24)(25,30)(26,34)(27,35)(28,36)(29,37)(31,39)(32,40)(33,38)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=h,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8835144);