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 | |
4263 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,7)(3,6,8,5)(9,10)(11,14)(12,20)(13,16)(15,22)(17,19)(18,24)(21,23)(26,32)(28,38)(33,35)(36,37);(1,2)(3,5)(4,7)(6,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,24)(20,22)(21,23)(25,26)(27,33)(28,34)(29,35)(30,36)(31,32)(37,40)(38,39);(1,3)(2,5)(4,8)(6,7)(9,10,13,16)(11,14,19,17)(12,15,22,20)(18,23,21,24)(25,27,31,29)(26,33,32,35)(28,37,38,36)(30,34,40,39);(1,3,4,8)(2,5,7,6)(9,10)(11,17)(12,15)(13,16)(14,19)(18,24)(20,22)(21,23)(25,28)(26,34)(27,37)(29,36)(30,35)(31,38)(32,39)(33,40);(1,3)(2,6)(4,8)(5,7)(9,11,13,19)(10,14,16,17)(12,21,22,18)(15,24,20,23)(25,29,31,27)(26,35,32,33)(28,37,38,36)(30,34,40,39);(1,2)(3,5)(4,7)(6,8)(9,12)(10,15)(11,18)(13,22)(14,23)(16,20)(17,24)(19,21)(25,30,31,40)(26,36,32,37)(27,34,29,39)(28,35,38,33)Polycyclic presentation:
2^9|a^2=hi,c^2=gh,d^2=hi,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10109270);