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 | |
4462 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)(4,5)(6,8)(9,10,14,15)(11,17,18,13)(12,20,21,16)(19,22,24,23)(26,27)(31,33)(32,34)(38,39);(2,5)(6,7)(10,15)(11,18)(12,21)(22,23)(25,26,30,27)(28,33,36,31)(29,32,37,34)(35,39,40,38);(1,2)(3,6)(4,5)(7,8)(10,15)(13,17)(16,20)(22,23)(29,37)(32,34)(35,40)(38,39);(1,3)(2,6)(4,8)(5,7)(9,11)(10,13)(12,19)(14,18)(15,17)(16,22)(20,23)(21,24)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,39,40,38);(9,12,14,21)(10,16,15,20)(11,19,18,24)(13,22,17,23)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39);(9,13)(10,11)(12,22)(14,17)(15,18)(16,19)(20,24)(21,23)(25,29)(26,32)(27,34)(28,35)(30,37)(31,38)(33,39)(36,40)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,d^2=ghi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8134539);