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 | |
4759 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(11,17)(15,18)(19,23)(22,24)(25,26)(27,33)(28,37)(29,38)(30,32)(31,36)(34,40)(35,39);(2,3)(6,7)(9,10)(11,15)(12,20)(13,14)(16,21)(17,18)(19,24)(22,23)(25,27,30,36)(26,31,32,33)(28,34,38,39)(29,40,37,35);(1,3,5,2)(4,7,8,6)(10,14)(15,18)(16,20)(22,24)(26,32)(28,38)(31,33)(34,39);(1,4)(2,6)(3,7)(5,8)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,28)(26,29)(27,34)(30,38)(31,40)(32,37)(33,35)(36,39);(9,11,13,17)(10,15,14,18)(12,19,21,23)(16,22,20,24)(25,27)(26,31)(28,39)(29,35)(30,36)(32,33)(34,38)(37,40);(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,29)(26,28)(27,35)(30,37)(31,39)(32,38)(33,34)(36,40)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=gh,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8762658);