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 | |
4471 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,14)(12,16)(13,17)(15,18)(19,22)(20,21)(23,24)(25,26,31,30)(27,32,37,36)(28,33,35,38)(29,34,40,39);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,19)(13,18)(15,17)(16,22)(20,23)(21,24)(25,27,31,37)(26,32,30,36)(28,29,35,40)(33,34,38,39);(1,3,5,4)(2,7,6,8)(11,18)(14,15)(19,24)(22,23)(26,30)(32,36)(33,38)(34,39);(1,4,5,3)(2,8,6,7)(9,10)(11,14)(12,20)(13,17)(15,18)(16,21)(19,23)(22,24)(25,28)(26,33)(27,29)(30,38)(31,35)(32,34)(36,39)(37,40);(9,11)(10,15)(12,19)(13,18)(14,17)(16,23)(20,22)(21,24)(25,29)(26,34)(27,35)(28,37)(30,39)(31,40)(32,38)(33,36);(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,20)(18,24)(25,30,31,26)(27,36,37,32)(28,38,35,33)(29,39,40,34)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,d^2=g,f^2=h,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7978695);