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 | |
4495 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(25,26,30,32)(27,35,37,31)(28,33,39,38)(29,36,40,34);(3,7)(5,8)(9,10,14,15)(11,12,16,20)(13,17,21,22)(18,19,23,24)(25,27,30,37)(26,31,32,35)(28,34,39,36)(29,33,40,38);(1,2)(3,5)(4,6)(7,8)(10,15)(11,16)(13,21)(19,24)(26,32)(29,40)(31,35)(33,38);(9,11)(10,12)(13,18)(14,16)(15,20)(17,19)(21,23)(22,24)(25,26)(27,31)(28,38)(29,34)(30,32)(33,39)(35,37)(36,40);(1,3)(2,5)(4,7)(6,8)(9,12)(10,16)(11,15)(13,19)(14,20)(17,23)(18,22)(21,24)(25,28,30,39)(26,33,32,38)(27,36,37,34)(29,35,40,31);(1,2)(3,5)(4,6)(7,8)(9,13)(10,17)(11,18)(12,19)(14,21)(15,22)(16,23)(20,24)(25,29)(26,34)(27,33)(28,35)(30,40)(31,39)(32,36)(37,38)Polycyclic presentation:
2^9|a^2=h,b^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8815564);