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 | |
4646 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(5,8)(9,10)(11,15)(12,18)(13,17)(14,16)(19,22)(20,21)(23,24)(25,26,30,34)(27,29,33,31)(28,32,36,39)(35,38,40,37);(1,2,4,6)(3,5,7,8)(9,11,14,18)(10,15,16,12)(13,19,23,21)(17,22,24,20)(25,27)(26,31)(28,35)(29,34)(30,33)(32,37)(36,40)(38,39);(1,3)(2,5)(4,7)(6,8)(10,16)(11,18)(17,24)(19,21)(28,36)(32,39)(35,40)(37,38);(1,2)(3,5)(4,6)(7,8)(9,11)(10,12)(13,21)(14,18)(15,16)(17,22)(19,23)(20,24)(25,28,30,36)(26,32,34,39)(27,35,33,40)(29,38,31,37);(1,3)(2,5)(4,7)(6,8)(9,12)(10,11)(13,22)(14,15)(16,18)(17,21)(19,24)(20,23)(25,27)(26,29)(28,35)(30,33)(31,34)(32,38)(36,40)(37,39);(9,13)(10,17)(11,19)(12,20)(14,23)(15,22)(16,24)(18,21)(25,29)(26,33)(27,34)(28,37)(30,31)(32,35)(36,38)(39,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=g,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, 8634035);