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 | |
4394 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10)(11,12)(13,21)(14,15)(16,19)(17,22)(18,23)(20,24)(25,26,31,29)(27,34,33,30)(28,36,37,32)(35,38,39,40);(3,4)(6,7)(10,15)(12,16)(17,21)(20,23)(26,29)(27,33)(28,37)(38,40);(1,2)(3,6)(4,7)(5,8)(25,27)(26,30)(28,35)(29,34)(31,33)(32,38)(36,40)(37,39);(1,3,5,4)(2,6,8,7)(9,11)(10,16)(12,15)(13,18)(14,19)(17,23)(20,21)(22,24)(25,28,31,37)(26,32,29,36)(27,35,33,39)(30,38,34,40);(1,4,5,3)(2,7,8,6)(9,12,14,16)(10,11,15,19)(13,20,22,23)(17,18,21,24)(25,29,31,26)(27,34,33,30)(28,36,37,32)(35,40,39,38);(1,2)(3,6)(4,7)(5,8)(9,13,14,22)(10,17,15,21)(11,18,19,24)(12,20,16,23)(25,30,31,34)(26,33,29,27)(28,38,37,40)(32,39,36,35)Polycyclic presentation:
2^9|a^2=h,d^2=h,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000545);