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 | |
4433 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,17,18,12)(13,21,23,16)(19,20,24,22)(25,26,31,30)(27,33,28,32)(29,34,37,38)(35,40,36,39);(1,2,5,8)(3,4,7,6)(10,15)(12,17)(16,21)(20,22)(25,27,31,28)(26,32,30,33)(29,35,37,36)(34,39,38,40);(1,3)(2,6)(4,8)(5,7)(10,15)(11,18)(16,21)(19,24);(9,11)(10,12)(13,19)(14,18)(15,17)(16,20)(21,22)(23,24)(25,28,31,27)(26,33,30,32)(29,36,37,35)(34,40,38,39);(1,3)(2,4)(5,7)(6,8)(9,12)(10,11)(13,22)(14,17)(15,18)(16,24)(19,21)(20,23)(25,29)(26,34)(27,35)(28,36)(30,38)(31,37)(32,39)(33,40);(1,4)(2,7)(3,8)(5,6)(9,13,14,23)(10,16,15,21)(11,19,18,24)(12,20,17,22)(25,30,31,26)(27,33,28,32)(29,34,37,38)(35,39,36,40)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9582105);