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 | |
4368 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,14)(11,17,12,16)(13,18,21,22)(19,24,20,23)(25,26)(27,29)(28,31)(30,36)(32,34)(33,38)(35,39)(37,40);(1,2)(3,8)(4,6)(5,7)(9,11,15,12)(10,16,14,17)(13,19,21,20)(18,23,22,24)(25,27)(26,32)(28,29)(30,35)(31,34)(33,39)(36,37)(38,40);(1,3,5,6)(2,4,7,8)(26,28)(27,34)(33,36)(35,40);(9,12,15,11)(10,17,14,16)(13,20,21,19)(18,24,22,23)(25,28,31,26)(27,29,34,32)(30,36,38,33)(35,37,40,39);(1,2)(3,4)(5,7)(6,8)(9,13)(10,18)(11,19)(12,20)(14,22)(15,21)(16,23)(17,24)(25,29)(26,27)(28,34)(30,37)(31,32)(33,35)(36,40)(38,39);(1,4)(2,6)(3,7)(5,8)(9,14,15,10)(11,17,12,16)(13,18,21,22)(19,23,20,24)(25,30,31,38)(26,33,28,36)(27,35,34,40)(29,37,32,39)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=hi,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9588912);