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 | |
4419 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)(9,10,14,15)(11,12,16,20)(13,21,22,17)(18,24,23,19)(25,26)(27,36)(28,39)(29,30)(31,32)(33,37)(34,38)(35,40);(1,3,4,7)(2,5,6,8)(9,11,14,16)(10,12,15,20)(13,18,22,23)(17,19,21,24)(26,32)(27,37)(29,35)(34,39);(1,2)(3,5)(4,6)(7,8)(10,15)(11,16)(13,22)(19,24)(25,27)(26,33)(28,35)(29,38)(30,39)(31,37)(32,36)(34,40);(9,10)(11,12)(13,17)(14,15)(16,20)(18,19)(21,22)(23,24)(25,28,31,38)(26,34,32,39)(27,35,37,29)(30,33,40,36);(1,3)(2,5)(4,7)(6,8)(9,12)(10,16)(11,15)(13,19)(14,20)(17,23)(18,21)(22,24)(25,29,31,35)(26,30,32,40)(27,38,37,28)(33,39,36,34);(1,2)(3,5)(4,6)(7,8)(9,13)(10,17)(11,18)(12,19)(14,22)(15,21)(16,23)(20,24)(25,30)(26,35)(27,39)(28,33)(29,32)(31,40)(34,37)(36,38)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8834158);