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 | |
4530 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)(25,26,30,34)(27,32,28,31)(29,33,37,40)(35,39,36,38);(2,3)(6,7)(9,10)(11,13)(12,17)(14,15)(16,19)(18,20)(21,22)(23,24)(25,27,30,28)(26,31,34,32)(29,35,37,36)(33,38,40,39);(1,3,5,2)(4,7,8,6)(10,15)(13,16)(17,22)(20,23)(25,27,30,28)(26,31,34,32)(29,36,37,35)(33,39,40,38);(1,4,5,8)(2,6,3,7)(9,11)(10,16)(12,18)(13,15)(14,19)(17,23)(20,22)(21,24)(25,28,30,27)(26,32,34,31)(29,36,37,35)(33,39,40,38);(9,12,14,21)(10,17,15,22)(11,18,19,24)(13,20,16,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(9,13,14,16)(10,11,15,19)(12,20,21,23)(17,18,22,24)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,d^2=g,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8104375);