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 | |
4738 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,17)(12,18)(13,21)(14,15)(16,22)(19,20)(23,24)(25,26)(27,34)(28,29)(30,31)(32,35)(33,39)(36,37)(38,40);(1,2,4,6)(3,7,8,5)(10,15)(12,17)(13,22)(19,24)(26,31)(29,33)(32,34)(37,40);(1,3)(2,5)(4,8)(6,7)(10,15)(11,18)(16,21)(19,24)(27,35)(32,34)(36,38)(37,40);(3,8)(5,7)(9,11,14,18)(10,12,15,17)(13,19,22,24)(16,20,21,23)(25,27)(26,32)(28,38)(29,37)(30,35)(31,34)(33,40)(36,39);(1,3)(2,5)(4,8)(6,7)(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,24)(22,23)(25,28)(26,33)(27,36)(29,31)(30,39)(32,37)(34,40)(35,38);(1,2)(3,5)(4,6)(7,8)(9,13,14,22)(10,16,15,21)(11,19,18,24)(12,20,17,23)(25,29,30,33)(26,28,31,39)(27,37,35,40)(32,38,34,36)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=hi,f^2=g,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, 8751085);