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 | |
4555 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,12)(13,17)(14,15)(16,19)(18,20)(21,22)(23,24);(3,7)(6,8)(10,15)(12,16)(17,22)(20,23)(25,26)(27,30)(28,32)(29,37)(31,34)(33,39)(35,40)(36,38);(1,2)(3,6)(4,5)(7,8)(13,21)(17,22)(18,24)(20,23)(25,27,31,28)(26,32,34,30)(29,35,39,36)(33,40,37,38);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,15)(13,18)(14,19)(17,23)(20,22)(21,24)(25,28,31,27)(26,32,34,30)(29,36,39,35)(33,40,37,38);(1,2)(3,6)(4,5)(7,8)(9,12,14,16)(10,11,15,19)(13,20,21,23)(17,18,22,24)(25,29,31,39)(26,33,34,37)(27,35,28,36)(30,38,32,40);(9,13,14,21)(10,17,15,22)(11,18,19,24)(12,20,16,23)(25,30,31,32)(26,27,34,28)(29,38,39,40)(33,35,37,36)Polycyclic presentation:
2^9|c^2=i,d^2=hi,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8174801);