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 | |
4722 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10,14,18)(11,19,21,15)(12,22,20,16)(13,17,23,24)(25,26)(27,34)(28,32)(29,39)(30,40)(31,33)(35,36)(37,38);(1,3,5,8)(2,6,7,4)(9,11,14,21)(10,15,18,19)(12,13,20,23)(16,17,22,24)(27,28)(32,34)(35,37)(36,38);(1,3)(2,4)(5,8)(6,7)(25,27,31,28)(26,32,33,34)(29,35,40,37)(30,36,39,38);(1,4)(2,3)(5,6)(7,8)(9,12)(10,16)(11,13)(14,20)(15,17)(18,22)(19,24)(21,23)(25,28,31,27)(26,32,33,34)(29,35,40,37)(30,38,39,36);(9,13)(10,17)(11,20)(12,21)(14,23)(15,22)(16,19)(18,24)(25,29)(26,30)(27,35)(28,37)(31,40)(32,36)(33,39)(34,38);(9,10)(11,15)(12,16)(13,17)(14,18)(19,21)(20,22)(23,24)(25,30)(26,29)(27,36)(28,38)(31,39)(32,35)(33,40)(34,37)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8042864);