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 | |
4443 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(10,12)(15,19)(16,20)(23,24)(25,26,31,34)(27,32,28,30)(29,33,38,40)(35,39,36,37);(1,2,5,6)(3,7,4,8)(9,10,14,12)(11,15,18,19)(13,16,22,20)(17,23,21,24)(25,27,31,28)(26,32,34,30)(29,35,38,36)(33,39,40,37);(10,12)(15,19)(16,20)(23,24)(25,27,31,28)(26,30,34,32)(29,35,38,36)(33,37,40,39);(1,3,5,4)(2,7,6,8)(9,11,14,18)(10,15,12,19)(13,21,22,17)(16,24,20,23)(25,28,31,27)(26,30,34,32)(29,35,38,36)(33,39,40,37);(1,3,5,4)(2,7,6,8)(9,12,14,10)(11,15,18,19)(13,20,22,16)(17,23,21,24)(25,29,31,38)(26,33,34,40)(27,35,28,36)(30,37,32,39);(1,4,5,3)(2,8,6,7)(9,13,14,22)(10,16,12,20)(11,17,18,21)(15,23,19,24)(25,30)(26,27)(28,34)(29,37)(31,32)(33,35)(36,40)(38,39)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=h,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7978544);