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 | |
4757 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)(9,10)(11,17)(12,20)(13,14)(15,18)(16,22)(19,23)(21,24)(25,26,30,32)(27,28,35,31)(29,37,39,33)(34,40,38,36);(2,3)(6,7)(10,14)(15,17)(16,20)(23,24)(25,27,30,35)(26,31,32,28)(29,38,39,34)(33,36,37,40);(1,3,5,2)(4,7,8,6)(11,18)(15,17)(19,21)(23,24)(26,32)(28,31)(33,37)(36,40);(1,4)(2,6)(3,7)(5,8)(9,11,13,18)(10,15,14,17)(12,21,22,19)(16,24,20,23)(25,28)(26,27)(29,36)(30,31)(32,35)(33,34)(37,38)(39,40);(9,12)(10,16)(11,19)(13,22)(14,20)(15,23)(17,24)(18,21)(25,27)(26,31)(28,32)(29,34)(30,35)(33,40)(36,37)(38,39);(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,22)(21,24)(25,29)(26,33)(27,34)(28,36)(30,39)(31,40)(32,37)(35,38)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=i,d^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, 8757781);