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 | |
4376 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)(3,7)(4,5)(6,8)(9,10,14,18)(11,16,12,15)(13,17,21,24)(19,23,20,22)(27,32)(29,34)(30,38)(33,36);(2,5)(6,7)(9,11,14,12)(10,15,18,16)(13,19,21,20)(17,22,24,23)(25,26,31,28)(27,34,32,29)(30,33,38,36)(35,40,39,37);(1,2)(3,6)(4,5)(7,8)(25,27)(26,29)(28,34)(30,35)(31,32)(33,37)(36,40)(38,39);(1,3,4,8)(2,6,5,7)(9,12,14,11)(10,16,18,15)(13,19,21,20)(17,22,24,23)(25,28,31,26)(27,34,32,29)(30,33,38,36)(35,37,39,40);(9,10)(11,16)(12,15)(13,17)(14,18)(19,23)(20,22)(21,24)(25,29,31,34)(26,32,28,27)(30,37,38,40)(33,39,36,35);(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,30,31,38)(26,33,28,36)(27,35,32,39)(29,37,34,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10329704);