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 | |
4365 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,15)(12,17)(13,21)(14,16)(18,23)(19,20)(22,24)(25,26,31,33)(27,36,37,32)(28,38,35,30)(29,34,39,40);(2,3)(6,7)(9,11,14,17)(10,15,16,12)(13,19,23,24)(18,22,21,20)(25,27)(26,32)(28,29)(30,34)(31,37)(33,36)(35,39)(38,40);(1,2,5,3)(4,6,8,7)(10,16)(11,17)(18,21)(19,24)(26,33)(27,37)(28,35)(34,40);(9,10)(11,15)(12,17)(13,18)(14,16)(19,22)(20,24)(21,23)(25,28,31,35)(26,30,33,38)(27,29,37,39)(32,34,36,40);(1,3,5,2)(4,6,8,7)(9,12,14,15)(10,17,16,11)(13,22,23,20)(18,19,21,24)(25,29)(26,34)(27,35)(28,37)(30,36)(31,39)(32,38)(33,40);(1,4)(2,6)(3,7)(5,8)(9,13)(10,18)(11,19)(12,20)(14,23)(15,22)(16,21)(17,24)(25,30)(26,35)(27,34)(28,33)(29,36)(31,38)(32,39)(37,40)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=hi,d^2=g,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9755154);