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 | |
4319 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(25,26)(27,35)(28,29)(30,40)(31,33)(32,36)(34,37)(38,39);(1,2)(3,5)(4,6)(7,8)(9,10,14,15)(11,16,20,23)(12,13,17,22)(18,24,21,19)(25,27)(26,32)(28,34)(29,30)(31,36)(33,35)(37,39)(38,40);(1,3,4,7)(2,5,6,8)(10,15)(13,22)(16,23)(19,24)(26,33)(27,36)(28,38)(30,37);(9,11)(10,16)(12,21)(13,19)(14,20)(15,23)(17,18)(22,24)(25,28,31,38)(26,29,33,39)(27,34,36,40)(30,35,37,32);(9,12,14,17)(10,13,15,22)(11,18,20,21)(16,24,23,19)(25,29)(26,28)(27,37)(30,36)(31,39)(32,40)(33,38)(34,35);(9,13)(10,17)(11,19)(12,15)(14,22)(16,18)(20,24)(21,23)(25,30,31,37)(26,34,33,40)(27,29,36,39)(28,35,38,32)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=gh,d^2=h,e^2=hi,f^2=h,b^a=bg,c^a=ch,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fi,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7919217);