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 | |
4761 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,14,15)(11,17,18,12)(13,21,22,16)(19,20,24,23)(25,26,30,33)(27,28,31,34)(29,32,38,40)(35,36,39,37);(2,3)(6,7)(9,11,14,18)(10,12,15,17)(13,19,22,24)(16,20,21,23)(25,27)(26,28)(29,35)(30,31)(32,36)(33,34)(37,40)(38,39);(1,3,5,2)(4,7,8,6)(10,15)(11,18)(16,21)(19,24)(27,31)(28,34)(35,39)(36,37);(1,4)(2,6)(3,7)(5,8)(9,10)(11,12)(13,16)(14,15)(17,18)(19,20)(21,22)(23,24)(25,26)(27,34)(28,31)(29,32)(30,33)(35,37)(36,39)(38,40);(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,24)(22,23)(25,28,30,34)(26,31,33,27)(29,37,38,36)(32,35,40,39);(9,13)(10,16)(11,19)(12,20)(14,22)(15,21)(17,23)(18,24)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,40)(34,37)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=gh,e^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9962889);