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 | |
4560 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,13)(11,17,18,15)(12,16,20,22)(19,24,21,23)(25,26,31,32)(27,30,35,38)(28,33,37,39)(29,34,36,40);(1,2)(3,7)(4,5)(6,8)(10,13)(11,18)(16,22)(19,21)(25,27,31,35)(26,30,32,38)(28,36,37,29)(33,40,39,34);(2,5)(6,7)(12,20)(16,22)(19,21)(23,24)(26,32)(27,35)(29,36)(33,39);(2,5)(6,7)(9,11,14,18)(10,15,13,17)(12,21,20,19)(16,24,22,23)(25,28)(26,33)(27,29)(30,34)(31,37)(32,39)(35,36)(38,40);(1,2)(3,6)(4,5)(7,8)(9,12,14,20)(10,16,13,22)(11,19,18,21)(15,23,17,24)(25,29)(26,34)(27,28)(30,33)(31,36)(32,40)(35,37)(38,39);(1,3)(2,6)(4,8)(5,7)(9,13,14,10)(11,17,18,15)(12,22,20,16)(19,24,21,23)(25,30)(26,27)(28,34)(29,33)(31,38)(32,35)(36,39)(37,40)Polycyclic presentation:
2^9|a^2=gh,b^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10227883);