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 | |
4719 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)(9,10,14,18)(11,15,19,16)(12,13,17,23)(20,21,24,22)(25,26)(27,28)(29,39)(30,35)(31,33)(32,37)(34,36)(38,40);(1,2,5,4)(3,7,8,6)(9,11)(10,15)(12,20)(13,21)(14,19)(16,18)(17,24)(22,23)(25,27,31,37)(26,32,33,28)(29,30,36,40)(34,35,39,38);(1,3,5,8)(2,6,4,7)(11,19)(15,16)(20,24)(21,22)(26,33)(28,32)(34,39)(35,38);(1,4,5,2)(3,7,8,6)(9,11)(10,16)(12,20)(13,22)(14,19)(15,18)(17,24)(21,23)(25,28,31,32)(26,27,33,37)(29,38,36,35)(30,39,40,34);(9,12)(10,13)(11,20)(14,17)(15,21)(16,22)(18,23)(19,24)(25,29)(26,34)(27,30)(28,38)(31,36)(32,35)(33,39)(37,40);(9,13)(10,17)(11,21)(12,18)(14,23)(15,24)(16,20)(19,22)(25,30)(26,35)(27,36)(28,34)(29,37)(31,40)(32,39)(33,38)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8638019);