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 | |
4830 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)(11,16)(12,22)(13,23)(14,17)(15,21)(18,24)(19,20)(25,26,30,36)(27,34,38,31)(28,33,39,32)(29,37,40,35);(9,11)(10,15)(12,19)(13,20)(14,21)(16,17)(18,23)(22,24)(25,27)(26,31)(28,35)(29,32)(30,38)(33,40)(34,36)(37,39);(1,2,5,4)(3,6,7,8)(9,12)(10,13)(11,18)(14,23)(15,24)(16,20)(17,22)(19,21)(25,28)(26,32)(27,35)(29,31)(30,39)(33,36)(34,40)(37,38);(3,7)(6,8)(9,12)(10,13)(11,19)(14,23)(15,20)(16,24)(17,22)(18,21)(25,28)(26,33)(27,37)(29,31)(30,39)(32,36)(34,40)(35,38);(1,3)(2,6)(4,8)(5,7)(9,11)(10,16)(12,18)(13,20)(14,21)(15,17)(19,23)(22,24)(25,27)(26,34)(28,35)(29,33)(30,38)(31,36)(32,40)(37,39);(1,4,5,2)(3,8,7,6)(9,13)(10,12)(11,20)(14,22)(15,19)(16,18)(17,23)(21,24)(25,29)(26,35)(27,32)(28,31)(30,40)(33,38)(34,39)(36,37)Polycyclic presentation:
2^9|a^2=gi,c^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10365166);