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 | |
4644 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(11,17)(15,22)(18,20)(23,24)(25,26,31,28)(27,35,36,32)(29,33,40,37)(30,34,39,38);(1,3,4,7)(2,5,6,8)(9,10)(11,15)(12,19)(13,14)(16,21)(17,22)(18,24)(20,23)(25,27,31,36)(26,32,28,35)(29,39,40,30)(33,34,37,38);(1,2)(3,5)(4,6)(7,8)(10,14)(15,22)(16,19)(23,24)(25,26,31,28)(27,32,36,35)(29,33,40,37)(30,38,39,34);(9,11)(10,15)(12,20)(13,17)(14,22)(16,24)(18,21)(19,23)(25,28,31,26)(27,35,36,32)(29,33,40,37)(30,38,39,34);(1,3)(2,5)(4,7)(6,8)(9,12)(10,16)(11,18)(13,21)(14,19)(15,23)(17,20)(22,24)(25,29)(26,33)(27,30)(28,37)(31,40)(32,38)(34,35)(36,39);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,22)(18,23)(19,21)(20,24)(25,30)(26,34)(27,29)(28,38)(31,39)(32,37)(33,35)(36,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=g,d^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8832108);