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 | |
4648 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(11,17)(15,22)(18,20)(23,24)(25,26,31,35)(27,32,37,34)(28,38,36,33)(29,30,39,40);(1,3,5,8)(2,6,7,4)(9,10)(11,15)(12,19)(13,14)(16,21)(17,22)(18,24)(20,23)(25,27)(26,32)(28,39)(29,36)(30,33)(31,37)(34,35)(38,40);(1,3)(2,4)(5,8)(6,7)(10,14)(15,22)(16,19)(23,24)(25,28,31,36)(26,33,35,38)(27,29,37,39)(30,32,40,34);(1,4)(2,3)(5,6)(7,8)(9,11)(10,15)(12,20)(13,17)(14,22)(16,24)(18,21)(19,23)(25,27)(26,34)(28,29)(30,33)(31,37)(32,35)(36,39)(38,40);(9,12)(10,16)(11,18)(13,21)(14,19)(15,23)(17,20)(22,24)(25,29)(26,30)(27,36)(28,37)(31,39)(32,33)(34,38)(35,40);(9,10)(11,15)(12,16)(13,14)(17,22)(18,23)(19,21)(20,24)(25,30)(26,29)(27,33)(28,34)(31,40)(32,36)(35,39)(37,38)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=h,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, 8832209);