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 | |
4640 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,12)(11,17,19,15)(13,16,21,20)(18,23,24,22)(25,26,30,31)(27,33,34,29)(28,36,37,32)(35,38,40,39);(1,2)(3,4)(5,6)(7,8)(10,12)(15,17)(16,20)(22,23)(26,31)(27,34)(28,37)(38,39);(2,6)(4,7)(13,21)(16,20)(18,24)(22,23)(26,31)(29,33)(32,36)(38,39);(9,11)(10,15)(12,17)(13,18)(14,19)(16,22)(20,23)(21,24)(25,27)(26,29)(28,35)(30,34)(31,33)(32,38)(36,39)(37,40);(1,3)(2,7)(4,6)(5,8)(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22)(25,28,30,37)(26,32,31,36)(27,35,34,40)(29,38,33,39);(1,4,5,7)(2,3,6,8)(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)(25,29)(26,27)(28,38)(30,33)(31,34)(32,35)(36,40)(37,39)Polycyclic presentation:
2^9|a^2=g,e^2=g,f^2=ghi,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, 8139490);