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 | |
4500 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,5)(3,7,8,6)(9,10,14,15)(11,19,21,16)(12,22,23,17)(13,18,24,20)(26,32)(27,28)(34,37)(35,36);(2,5)(6,7)(10,15)(12,23)(16,19)(18,20)(25,26)(27,33)(28,30)(29,37)(31,32)(34,39)(35,38)(36,40);(1,2)(3,6)(4,5)(7,8)(25,27,31,28)(26,33,32,30)(29,35,39,36)(34,40,37,38);(1,2)(3,6)(4,5)(7,8)(9,11,14,21)(10,16,15,19)(12,20,23,18)(13,22,24,17)(25,28,31,27)(26,33,32,30)(29,36,39,35)(34,40,37,38);(1,3,4,8)(2,6,5,7)(9,12,14,23)(10,17,15,22)(11,20,21,18)(13,19,24,16)(25,29)(26,34)(27,35)(28,36)(30,38)(31,39)(32,37)(33,40);(9,13)(10,18)(11,17)(12,19)(14,24)(15,20)(16,23)(21,22)(25,30,31,33)(26,28,32,27)(29,38,39,40)(34,36,37,35)Polycyclic presentation:
2^9|a^2=ghi,c^2=hi,d^2=gi,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8416133);