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 | |
4508 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)(6,7)(11,17)(12,20)(15,18)(16,21)(25,26)(27,33)(28,37)(29,39)(30,31)(32,36)(34,40)(35,38);(1,2)(3,6)(4,5)(7,8)(9,10)(11,18)(12,21)(13,14)(15,17)(16,20)(19,23)(22,24)(25,26)(27,33)(28,29)(30,31)(32,36)(34,35)(37,39)(38,40);(1,3,5,8)(2,6,4,7)(10,14)(15,18)(16,21)(23,24)(26,31)(29,37)(32,33)(35,40);(2,4)(6,7)(9,11)(10,15)(12,22)(13,17)(14,18)(16,24)(19,20)(21,23)(25,27,30,36)(26,32,31,33)(28,38,39,34)(29,35,37,40);(1,4,5,2)(3,7,8,6)(9,12)(10,16)(11,19)(13,20)(14,21)(15,23)(17,22)(18,24)(25,28)(26,29)(27,34)(30,39)(31,37)(32,40)(33,35)(36,38);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,21)(22,24)(25,29)(26,28)(27,35)(30,37)(31,39)(32,38)(33,34)(36,40)Polycyclic presentation:
2^9|c^2=g,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8839968);