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 | |
4372 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)(3,7)(4,5)(6,8)(9,10,14,15)(11,12,17,16)(13,21,22,18)(19,23,24,20)(25,26)(27,29)(28,30)(31,33)(32,36)(34,38)(35,39)(37,40);(2,5)(6,7)(9,11)(10,12)(13,19)(14,17)(15,16)(18,20)(21,23)(22,24)(25,27,31,36)(26,32,33,29)(28,34,37,39)(30,35,40,38);(1,2)(3,6)(4,5)(7,8)(10,15)(11,17)(13,22)(20,23)(28,37)(30,40)(34,39)(35,38);(1,3)(2,6)(4,8)(5,7)(9,11)(10,16)(12,15)(13,19)(14,17)(18,23)(20,21)(22,24)(25,28)(26,30)(27,34)(29,38)(31,37)(32,35)(33,40)(36,39);(9,12,14,16)(10,17,15,11)(13,20,22,23)(18,24,21,19)(25,29,31,32)(26,27,33,36)(28,35,37,38)(30,39,40,34);(9,13,14,22)(10,18,15,21)(11,19,17,24)(12,20,16,23)(25,30)(26,28)(27,35)(29,39)(31,40)(32,34)(33,37)(36,38)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,e^2=gi,f^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, 8839359);