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 | |
4420 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10)(11,15)(12,14)(13,20)(16,21)(17,18)(19,24)(22,23)(25,26,31,34)(27,33,30,32)(28,37,38,29)(35,36,39,40);(3,7)(6,8)(9,11,14,17)(10,15,12,18)(13,19,21,23)(16,22,20,24)(25,27,31,30)(26,32,34,33)(28,35,38,39)(29,36,37,40);(1,2,5,4)(3,6,7,8)(10,12)(11,17)(16,20)(19,23)(28,38)(29,37)(35,39)(36,40);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,28)(26,29)(27,35)(30,39)(31,38)(32,36)(33,40)(34,37);(1,4,5,2)(3,8,7,6)(9,12,14,10)(11,18,17,15)(13,20,21,16)(19,24,23,22)(25,29)(26,28)(27,36)(30,40)(31,37)(32,35)(33,39)(34,38);(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,23)(18,24)(25,30,31,27)(26,33,34,32)(28,39,38,35)(29,40,37,36)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8034625);