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 | |
4384 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,18)(12,15)(13,21)(14,17)(16,22)(19,20)(23,24)(25,26,31,28)(27,34,30,32)(29,33,38,36)(35,40,37,39);(1,2,4,6)(3,5,8,7)(9,11,14,15)(10,12,17,18)(13,19,22,23)(16,20,21,24)(25,26,31,28)(27,32,30,34)(29,36,38,33)(35,40,37,39);(1,2)(3,5)(4,6)(7,8)(9,10,14,17)(11,12,15,18)(13,21,22,16)(19,24,23,20)(25,27,31,30)(26,32,28,34)(29,35,38,37)(33,39,36,40);(1,3,4,8)(2,5,6,7)(9,10)(11,12)(13,16)(14,17)(15,18)(19,20)(21,22)(23,24)(25,28,31,26)(27,34,30,32)(29,36,38,33)(35,40,37,39);(9,12)(10,15)(11,17)(13,20)(14,18)(16,23)(19,21)(22,24)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40);(9,13,14,22)(10,16,17,21)(11,19,15,23)(12,20,18,24)(25,30,31,27)(26,34,28,32)(29,37,38,35)(33,40,36,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=h,d^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8837188);