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 | |
4456 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)(10,15)(11,17)(18,22)(19,24)(25,26)(27,32)(28,34)(29,35)(30,31)(33,37)(36,40)(38,39);(2,3)(6,7)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,24)(21,22)(25,27,31,39)(26,32,30,38)(28,36,37,29)(33,35,34,40);(1,3,5,2)(4,7,8,6)(11,17)(12,16)(13,21)(18,22)(25,28,31,37)(26,33,30,34)(27,29,39,36)(32,40,38,35);(1,4,5,8)(2,6,3,7)(9,11)(10,16)(12,15)(13,19)(14,17)(18,23)(20,22)(21,24)(25,28)(26,34)(27,36)(29,39)(30,33)(31,37)(32,40)(35,38);(9,12,14,16)(10,17,15,11)(13,20,21,23)(18,24,22,19)(25,29,31,36)(26,35,30,40)(27,37,39,28)(32,33,38,34);(9,13)(10,18)(11,19)(12,20)(14,21)(15,22)(16,23)(17,24)(25,30,31,26)(27,38,39,32)(28,33,37,34)(29,40,36,35)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=i,d^2=hi,e^2=gi,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, 8922544);