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 | |
4685 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,3)(4,8)(9,10)(11,18)(12,13)(14,15)(16,19)(17,23)(20,21)(22,24)(25,26)(27,31)(28,32)(29,38)(30,34)(33,39)(35,36)(37,40);(2,3)(4,8)(10,15)(13,17)(16,18)(21,24)(25,27)(26,31)(28,29)(30,35)(32,38)(33,40)(34,36)(37,39);(1,2,5,3)(4,6,8,7)(11,19)(16,18)(20,22)(21,24)(27,35)(28,37)(31,36)(32,40);(2,3)(6,7)(9,11,14,19)(10,16,15,18)(12,22,23,20)(13,21,17,24)(25,28,30,37)(26,32,34,40)(27,29,35,39)(31,38,36,33);(1,3,5,2)(4,7,8,6)(9,12)(10,17)(11,20)(13,15)(14,23)(16,21)(18,24)(19,22)(25,29)(26,33)(27,28)(30,39)(31,40)(32,36)(34,38)(35,37);(1,4,5,8)(2,6,3,7)(9,13,14,17)(10,12,15,23)(11,21,19,24)(16,22,18,20)(25,26)(27,36)(28,32)(29,33)(30,34)(31,35)(37,40)(38,39)Polycyclic presentation:
2^9|c^2=h,d^2=ghi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8746313);