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 | |
4672 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10,14,15)(11,18,17,13)(12,20,21,16)(19,22,23,24)(25,26)(27,31)(28,36)(29,37)(30,32)(33,38)(34,40)(35,39);(3,7)(6,8)(10,15)(11,17)(16,20)(19,23)(25,27,30,33)(26,31,32,38)(28,34,37,39)(29,35,36,40);(1,2)(3,6)(4,5)(7,8)(10,15)(11,17)(12,21)(22,24)(26,32)(27,33)(28,37)(35,40);(1,3)(2,6)(4,7)(5,8)(9,11,14,17)(10,13,15,18)(12,19,21,23)(16,22,20,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(1,2)(3,6)(4,5)(7,8)(9,12)(10,16)(11,19)(13,22)(14,21)(15,20)(17,23)(18,24)(25,28)(26,29)(27,34)(30,37)(31,35)(32,36)(33,39)(38,40);(9,13)(10,17)(11,15)(12,22)(14,18)(16,23)(19,20)(21,24)(25,29)(26,28)(27,35)(30,36)(31,34)(32,37)(33,40)(38,39)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8033276);