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 | |
4324 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,26)(27,38)(28,33)(29,39)(30,35)(31,36)(32,37)(34,40);(1,2,5,7)(3,4,8,6)(11,17)(15,21)(18,23)(22,24)(25,27)(26,28)(29,35)(30,34)(31,40)(32,33)(36,39)(37,38);(1,3)(2,6)(4,7)(5,8)(10,14)(15,21)(16,19)(22,24)(25,28,32,38)(26,33,37,27)(29,30,40,36)(31,39,35,34);(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,29,32,40)(26,34,37,39)(27,35,33,31)(28,30,38,36);(1,4,5,6)(2,3,7,8)(9,11,13,17)(10,15,14,21)(12,18,20,23)(16,22,19,24)(25,30)(26,35)(27,39)(28,40)(29,38)(31,37)(32,36)(33,34);(1,3)(2,4)(5,8)(6,7)(9,12,13,20)(10,16,14,19)(11,18,17,23)(15,22,21,24)(25,31,32,35)(26,36,37,30)(27,40,33,29)(28,39,38,34)Polycyclic presentation:
2^9|b^2=i,c^2=g,d^2=g,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9751332);