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 | |
4619 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,13,15)(11,14,17,22)(12,16,19,23)(18,21,20,24)(25,26)(27,29)(28,31)(30,36)(32,34)(33,38)(35,39)(37,40);(2,3)(6,7)(9,11,13,17)(10,14,15,22)(12,18,19,20)(16,21,23,24)(25,27)(26,32)(28,29)(30,35)(31,34)(33,39)(36,37)(38,40);(1,3,5,2)(4,7,8,6)(10,15)(11,17)(12,19)(21,24)(26,28)(27,34)(33,36)(35,40);(1,4,5,8)(2,6,3,7)(9,12)(10,16)(11,18)(13,19)(14,21)(15,23)(17,20)(22,24)(25,28,31,26)(27,29,34,32)(30,36,38,33)(35,37,40,39);(9,11)(10,14)(12,20)(13,17)(15,22)(16,24)(18,19)(21,23)(25,29)(26,27)(28,34)(30,37)(31,32)(33,35)(36,40)(38,39);(9,10)(11,14)(12,16)(13,15)(17,22)(18,21)(19,23)(20,24)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=hi,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9401889);