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 | |
4779 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,7)(9,10)(11,17)(12,18)(13,21)(14,15)(16,22)(19,20)(23,24)(25,26,30,33)(27,29,35,31)(28,36,37,32)(34,38,40,39);(1,2,5,4)(3,7,8,6)(10,15)(12,17)(13,22)(19,24)(25,27,30,35)(26,31,33,29)(28,34,37,40)(32,38,36,39);(1,3,5,8)(2,6,4,7)(10,15)(11,18)(16,21)(19,24)(28,37)(32,36)(34,40)(38,39);(1,4,5,2)(3,7,8,6)(9,11,14,18)(10,12,15,17)(13,19,22,24)(16,20,21,23)(25,28)(26,32)(27,34)(29,39)(30,37)(31,38)(33,36)(35,40);(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,24)(22,23)(25,29)(26,27)(28,38)(30,31)(32,40)(33,35)(34,36)(37,39);(9,13)(10,16)(11,19)(12,20)(14,22)(15,21)(17,23)(18,24)(25,27)(26,29)(28,34)(30,35)(31,33)(32,39)(36,38)(37,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=i,d^2=hi,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, 8757776);