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 | |
4478 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,17)(12,20)(13,14)(15,18)(16,21)(19,22)(23,24)(25,26)(27,35)(28,30)(29,32)(31,34)(33,40)(36,37)(38,39);(1,2)(3,6)(4,5)(7,8)(11,18)(15,17)(19,24)(22,23)(25,27,31,29)(26,32,34,35)(28,38,33,36)(30,37,40,39);(10,14)(15,17)(16,20)(22,23)(27,29)(32,35)(36,38)(37,39);(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,23)(21,24)(25,28,31,33)(26,30,34,40)(27,36,29,38)(32,39,35,37);(1,3,4,7)(2,6,5,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,29,31,27)(26,32,34,35)(28,38,33,36)(30,39,40,37);(9,12,13,21)(10,16,14,20)(11,19,18,24)(15,22,17,23)(25,30)(26,33)(27,37)(28,34)(29,39)(31,40)(32,36)(35,38)Polycyclic presentation:
2^9|b^2=i,d^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,d^b=dhi,e^c=ehi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7551238);