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 | |
4509 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)(3,8)(4,7)(5,6)(9,10,14,12)(11,17,18,15)(13,20,21,16)(19,22,24,23)(25,26,30,33)(27,34,28,31)(29,32,38,37)(35,40,36,39);(3,4)(7,8)(10,12)(15,17)(16,20)(22,23)(25,27,30,28)(26,31,33,34)(29,35,38,36)(32,39,37,40);(1,3)(2,7)(4,6)(5,8)(10,12)(11,18)(13,21)(22,23)(25,26,30,33)(27,31,28,34)(29,37,38,32)(35,40,36,39);(3,4)(7,8)(9,11,14,18)(10,15,12,17)(13,19,21,24)(16,22,20,23)(25,26)(27,31)(28,34)(29,32)(30,33)(35,39)(36,40)(37,38);(1,4,6,3)(2,7,5,8)(9,12,14,10)(11,15,18,17)(13,20,21,16)(19,22,24,23)(25,28,30,27)(26,31,33,34)(29,36,38,35)(32,39,37,40);(1,5,6,2)(3,7,4,8)(9,13,14,21)(10,16,12,20)(11,19,18,24)(15,22,17,23)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,37)(34,40)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=ghi,d^2=hi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8923843);