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 | |
4480 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10)(11,18)(12,16)(13,21)(14,15)(17,22)(19,20)(23,24);(2,6)(3,7)(10,15)(12,18)(17,21)(20,24)(25,26,30,31)(27,28,32,36)(29,33,37,38)(34,35,39,40);(1,3)(2,4)(5,7)(6,8)(10,15)(11,16)(13,22)(20,24)(26,31)(27,32)(29,37)(35,40);(1,4)(2,3)(5,8)(6,7)(9,11)(10,12)(13,19)(14,16)(15,18)(17,20)(21,24)(22,23)(25,27)(26,28)(29,34)(30,32)(31,36)(33,35)(37,39)(38,40);(9,12)(10,16)(11,15)(13,20)(14,18)(17,23)(19,21)(22,24)(25,28)(26,32)(27,31)(29,35)(30,36)(33,39)(34,38)(37,40);(9,13,14,22)(10,17,15,21)(11,19,16,23)(12,20,18,24)(25,29)(26,33)(27,34)(28,35)(30,37)(31,38)(32,39)(36,40)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8816971);