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 | |
4568 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)(3,4)(5,6)(7,8)(9,10,14,11)(12,15,19,17)(13,18,21,16)(20,23,24,22)(25,26,29,32)(27,33,35,30)(28,31,37,40)(34,39,36,38);(1,2)(3,8)(4,7)(5,6)(12,19)(15,17)(20,24)(22,23)(25,27)(26,30)(28,36)(29,35)(31,39)(32,33)(34,37)(38,40);(2,6)(3,7)(10,11)(15,17)(16,18)(22,23)(25,27,29,35)(26,30,32,33)(28,34,37,36)(31,38,40,39);(1,2,5,6)(3,4,7,8)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,37)(39,40);(1,3)(2,4)(5,7)(6,8)(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,28,29,37)(26,31,32,40)(27,34,35,36)(30,38,33,39);(1,4)(2,7)(3,6)(5,8)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,38)(36,39)(37,40)Polycyclic presentation:
2^9|a^2=ghi,c^2=g,d^2=gh,e^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9963815);