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 | |
4325 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)(25,26)(27,36)(28,30)(29,39)(31,35)(32,38)(33,37)(34,40);(3,7)(6,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,21)(22,24)(25,27,31,38)(26,32,35,36)(28,29,37,40)(30,34,33,39);(1,2,5,4)(3,6,7,8)(10,15)(13,23)(16,21)(17,24)(25,28,31,37)(26,33,35,30)(27,29,38,40)(32,39,36,34);(1,3)(2,6)(4,8)(5,7)(9,11)(10,16)(12,18)(13,17)(14,20)(15,21)(19,22)(23,24)(25,27)(26,32)(28,29)(30,34)(31,38)(33,39)(35,36)(37,40);(1,4,5,2)(3,8,7,6)(9,12)(10,17)(11,19)(13,21)(14,22)(15,24)(16,23)(18,20)(25,29)(26,34)(27,37)(28,38)(30,36)(31,40)(32,33)(35,39);(9,13,14,23)(10,18,15,19)(11,17,20,24)(12,21,22,16)(25,30,31,33)(26,28,35,37)(27,34,38,39)(29,36,40,32)Polycyclic presentation:
2^9|b^2=g,c^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8229501);