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 | |
4851 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,7)(9,10)(11,19)(12,21)(13,18)(14,15)(16,20)(17,22)(23,24)(25,26,31,34)(27,32,35,29)(28,30,33,39)(36,37,40,38);(1,2,5,4)(3,7,8,6)(10,15)(11,20)(17,21)(18,24)(25,27,31,35)(26,32,34,29)(28,36,33,40)(30,37,39,38);(1,3,5,8)(2,6,4,7)(10,15)(12,22)(16,19)(18,24)(27,35)(29,32)(36,40)(37,38);(1,4,5,2)(3,7,8,6)(9,11,14,20)(10,16,15,19)(12,18,22,24)(13,21,23,17)(25,28)(26,30)(27,36)(29,38)(31,33)(32,37)(34,39)(35,40);(9,12)(10,17)(11,18)(13,16)(14,22)(15,21)(19,23)(20,24)(25,29)(26,27)(28,38)(30,36)(31,32)(33,37)(34,35)(39,40);(9,13)(10,18)(11,17)(12,16)(14,23)(15,24)(19,22)(20,21)(25,30)(26,33)(27,37)(28,34)(29,36)(31,39)(32,40)(35,38)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8428834);