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 | |
4657 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)(9,10)(11,17)(12,20)(13,14)(15,18)(16,21)(19,22)(23,24)(26,31)(27,33)(32,39)(34,36);(2,3)(6,7)(10,14)(15,17)(16,20)(22,23)(25,26)(27,29)(28,32)(30,31)(33,38)(34,37)(35,39)(36,40);(1,3,5,2)(4,7,8,6)(10,14)(11,18)(12,21)(22,23)(27,33)(28,35)(29,38)(32,39);(1,4,5,8)(2,6,3,7)(9,11,13,18)(10,15,14,17)(12,19,21,24)(16,22,20,23)(25,27)(26,29)(28,36)(30,33)(31,38)(32,40)(34,35)(37,39);(9,10)(11,17)(12,16)(13,14)(15,18)(19,23)(20,21)(22,24)(25,28)(26,32)(27,34)(29,37)(30,35)(31,39)(33,36)(38,40);(9,12,13,21)(10,16,14,20)(11,19,18,24)(15,22,17,23)(25,29)(26,27)(28,37)(30,38)(31,33)(32,34)(35,40)(36,39)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8900704);