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 | |
4701 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)(3,7)(4,8)(5,6)(9,10)(11,17)(12,16)(13,14)(15,19)(18,24)(20,22)(21,23)(25,26,29,32)(27,33,34,30)(28,31,38,37)(35,40,36,39);(2,6)(3,8)(10,14)(15,17)(16,20)(23,24)(27,34)(30,33)(35,36)(39,40);(1,3)(2,4)(5,8)(6,7)(10,14)(15,17)(16,20)(23,24)(25,27,29,34)(26,30,32,33)(28,35,38,36)(31,39,37,40);(1,3)(2,4)(5,8)(6,7)(9,10)(11,15)(12,20)(13,14)(16,22)(17,19)(18,24)(21,23)(25,27)(26,30)(28,36)(29,34)(31,40)(32,33)(35,38)(37,39);(1,4)(2,3)(5,7)(6,8)(9,11)(10,15)(12,21)(13,19)(14,17)(16,24)(18,22)(20,23)(25,26)(27,30)(28,37)(29,32)(31,38)(33,34)(35,40)(36,39);(9,12)(10,16)(11,18)(13,22)(14,20)(15,23)(17,24)(19,21)(25,28,29,38)(26,31,32,37)(27,35,34,36)(30,39,33,40)Polycyclic presentation:
2^9|a^2=gh,c^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8636173);