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 | |
4599 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)(11,14)(12,17)(13,18)(15,20)(16,21)(19,22)(23,24)(25,26)(27,29)(28,33)(30,38)(31,32)(34,39)(35,40)(36,37);(3,7)(6,8)(10,11)(15,17)(16,18)(22,23)(25,27)(26,29)(28,31)(30,35)(32,33)(34,37)(36,39)(38,40);(1,2,5,4)(3,6,7,8)(10,11)(15,17)(16,18)(22,23)(26,32)(27,28)(34,38)(35,36);(1,3,5,7)(2,6,4,8)(9,11,14,10)(12,15,20,17)(13,18,21,16)(19,22,24,23)(25,28,31,27)(26,33,32,29)(30,36,39,35)(34,40,38,37);(1,4,5,2)(3,8,7,6)(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,29,31,33)(26,28,32,27)(30,37,39,40)(34,36,38,35);(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,30)(26,34)(27,35)(28,36)(29,37)(31,39)(32,38)(33,40)Polycyclic presentation:
2^9|c^2=ghi,d^2=h,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^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8064527);