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 | |
4752 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10,14,16)(11,15,18,12)(13,17,22,23)(19,21,24,20)(25,26)(27,35)(28,37)(29,31)(30,34)(32,36)(33,39)(38,40);(3,7)(6,8)(9,11)(10,15)(12,16)(13,19)(14,18)(17,21)(20,23)(22,24)(25,27,31,36)(26,32,29,35)(28,38,39,30)(33,40,37,34);(1,2,5,4)(3,6,7,8)(10,16)(11,18)(17,23)(19,24)(26,29)(32,35)(33,37)(34,40);(1,3,5,7)(2,6,4,8)(9,12)(10,11)(13,21)(14,15)(16,18)(17,24)(19,23)(20,22)(25,28,31,39)(26,33,29,37)(27,30,36,38)(32,34,35,40);(1,4,5,2)(3,8,7,6)(9,11)(10,12)(13,19)(14,18)(15,16)(17,20)(21,23)(22,24)(25,29,31,26)(27,35,36,32)(28,37,39,33)(30,40,38,34);(9,13,14,22)(10,17,16,23)(11,19,18,24)(12,20,15,21)(25,30)(26,34)(27,28)(29,40)(31,38)(32,33)(35,37)(36,39)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=ghi,d^2=hi,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8488673);