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 | |
4467 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,16)(12,18)(13,23)(14,19)(15,17)(20,22)(21,24);(1,3,5,8)(2,6,7,4)(9,11)(10,16)(12,17)(13,20)(14,21)(15,18)(19,24)(22,23)(25,26)(27,34)(28,31)(29,32)(30,33)(35,39)(36,37)(38,40);(1,3)(2,4)(5,8)(6,7)(10,17)(11,18)(13,19)(21,22)(25,27,30,32)(26,29,33,34)(28,36,38,35)(31,39,40,37);(1,4)(2,3)(5,6)(7,8)(9,12,15,16)(10,18,17,11)(13,22,19,21)(14,24,23,20)(25,26)(27,29)(28,31)(30,33)(32,34)(35,37)(36,39)(38,40);(9,13,15,19)(10,14,17,23)(11,20,18,24)(12,22,16,21)(25,28,30,38)(26,31,33,40)(27,35,32,36)(29,37,34,39);(9,14)(10,19)(11,21)(12,20)(13,17)(15,23)(16,24)(18,22)(25,29)(26,32)(27,33)(28,37)(30,34)(31,36)(35,40)(38,39)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=ghi,d^2=h,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8441644);